# Analytic Nullstellensätze
and the Model Theory of Valued Fields

Matthias Aschenbrenner  Kurt Gödel Research Center for Mathematical Logic
Universität Wien
1090 Wien
Austria<EMAIL_ADDRESS>and  Ahmed Srhir  Département de
Mathématique
Faculté des Sciences
Université Ibn Tofail
Kénitra
Morocco<EMAIL_ADDRESS>
(Date: February 2023)

###### Abstract.

We present a uniform framework for establishing Nullstellensätze for power
series rings using quantifier elimination results for valued fields. As an
application we obtain Nullstellensätze for $p$-adic power series (both formal
and convergent) analogous to Rückert’s complex and Risler’s real
Nullstellensatz, as well as a $p$-adic analytic version of Hilbert’s 17th
Problem. Analogous statements for restricted power series, both real and
$p$-adic, are also considered.

## Introduction

Rückert’s Nullstellensatz [56] establishes a one-to-one correspondence between
radical ideals in rings of germs of complex analytic functions and the zero
sets of these ideals. This fundamental theorem plays a role in complex
analytic geometry which is similar to that of Hilbert’s Nullstellensatz in
classical algebraic geometry. A counterpart of Hilbert’s Nullstellensatz for
polynomials over real closed fields (instead of algebraically closed fields)
is due to Risler [49], who also established a version of Rückert’s theorem for
germs of real analytic functions [50] as well as an analogue of Hilbert’s 17th
Problem in this setting. An adaptation of Risler’s theorem for formal power
series over real closed fields was shown by Merrien [43, 44] (with
simplifications in [38]) and used for applications to germs of real
$C^{\infty}$-functions.

It was noted early in the history of model theory that there is a close
relationship between Nullstellensatz-type statements and A. Robinson’s concept
of model completeness: for example, Hilbert’s Nullstellensatz is essentially
equivalent to the model completeness of the theory of algebraically closed
fields, whereas Risler’s Nullstellensatz for polynomials corresponds to the
model completeness of the theory of real closed fields. (This connection has
been formalized in [62].)

A. Robinson [55] established Rückert’s Nullstellensatz using non-standard
methods, and asked for a model-theoretic proof. This was provided by
Weispfenning [61], who also gave a proof of Risler’s theorem for real analytic
function germs using similar methods. His arguments, which were translated
into somewhat more algebraic language by Robbin [52, 53], combine the
ubiquitous Weierstrass Preparation Theorem (which permits an induction on the
number of indeterminates) with model completeness of algebraically closed
fields respectively real closed fields.

In this paper we establish a uniform method for obtaining Nullstellensätze in
power series rings by relying on quantifier elimination theorems for certain
theories of valued fields instead of model completeness for fields in order to
prove the requisite specialization theorems. The advantage of this approach is
that it naturally suggests how to also prove Nullstellensätze and versions of
Hilbert’s 17th Problem for $p$-adic power series, which seem to be new. (See
[24, Conjectures 4.3, 4.4].)

### Nullstellensätze for power series over the $p$-adics

To formulate these, consider the rational function

$\gamma_{p}(x):=\frac{1}{p}\left(\wp(x)-\frac{1}{\wp(x)}\right)^{-1}\in{\mathbb{Q}}(x)$

where $\wp(x):=x^{p}-x$ (the Artin-Schreier operator). One calls
$\gamma=\gamma_{p}$ the $p$-adic Kochen operator. It plays a role in $p$-adic
algebra similar to that of the squaring operator $x^{2}$ in the real world. To
substantiate this, we remark that $\gamma$ is $p$-integral definite, that is,
$\lvert\gamma(a)\rvert_{p}\leqslant 1$ for all $a\in{\mathbb{Q}}_{p}$. (Here
and below $\lvert\,\cdot\,\rvert_{p}$ denotes the $p$-adic absolute value on
${\mathbb{Q}}_{p}$.) As a consequence, setting $F:={\mathbb{Q}}_{p}(X)$ where
$X=(X_{1},\dots,X_{m})$ is a tuple of distinct indeterminates, every rational
function of the form $\gamma(f)$ where $f\in F$ is $p$-integral definite, and
hence so is every element of

$\Lambda_{F}:=\left\\{\frac{f}{1-pg}:f,g\in{\mathbb{Z}}[\gamma(F)]\right\\}.$

Kochen [34] proved that conversely, the subring $\Lambda_{F}$ of $F$ contains
every $p$-integral definite rational function in $F$; this may be seen as a
$p$-adic analogue of Hilbert’s 17th Problem. Kochen also established a
$p$-adic Nullstellensatz for ideals in ${\mathbb{Q}}_{p}[X]$, similar to
Risler’s Nullstellensatz for ideals in ${\mathbb{R}}[X]$, the statement of
which also involves the ring $\Lambda_{F}$. Our first main theorem is an
analogous result for ideals in the ring ${\mathbb{Q}}_{p}\\{X\\}$ of
convergent power series over ${\mathbb{Q}}_{p}$ in the indeterminates $X$. In
the following, we let $A$ be a subring of ${\mathbb{Q}}_{p}[[X]]$ with
fraction field $F=\operatorname{Frac}(A)$. Then $1\neq pg$ for each
$g\in{\mathbb{Z}}[\gamma(F)]$, and so we may define the subring $\Lambda_{F}$
of $F$ as above. We also let $\Lambda_{F}A$ denote the subring of $F$
generated by $\Lambda_{F}$ and $A$. With these notations, here is our $p$-adic
analytic Nullstellensatz:

######  Theorem A.

Suppose $A={\mathbb{Q}}_{p}\\{X\\}$, and let $f,g_{1},\dots,g_{n}\in A$. If

$g_{1}(a)=\cdots=g_{n}(a)=0\ \Longrightarrow\ f(a)=0\quad\text{for each
$a\in{\mathbb{Q}}_{p}^{m}$ sufficiently close to $0$,}$

then there are $h_{1},\dots,h_{n}\in\Lambda_{F}A$ and $k\geqslant 1$ such that
$f^{k}=g_{1}h_{1}+\cdots+g_{n}h_{n}$.

The converse of the implication in this theorem holds trivially. Next, an
analytic version of Kochen’s $p$-adic Hilbert’s 17th Problem:

######  Theorem B.

Suppose $A={\mathbb{Q}}_{p}\\{X\\}$, and let $f,g\in A$. Then $\lvert
f(a)\rvert_{p}\leqslant\lvert g(a)\rvert_{p}$ for all
$a\in{\mathbb{Q}}_{p}^{m}$ sufficiently close to $0$ if and only if $f\in
g\Lambda_{F}$.

We also have a $p$-adic version of Merrien’s theorem for formal power series,
where we now let $t$ be a single indeterminate over ${\mathbb{Q}}_{p}$:

######  Theorem C.

Suppose $A={\mathbb{Q}}_{p}[[X]]$, and let $f,g_{1},\dots,g_{n}\in A$ be such
that

$g_{1}(a)=\cdots=g_{n}(a)=0\ \Longrightarrow\ f(a)=0\quad\text{for each
$a\in{\mathbb{Q}}_{p}[[t]]^{m}$ with $a(0)=0$.}$

Then there are $h_{1},\dots,h_{n}\in\Lambda_{F}A$ and $k\geqslant 1$ such that
$f^{k}=g_{1}h_{1}+\cdots+g_{n}h_{n}$.

In our arguments we systematically work in the framework of Weierstrass
systems from [17], which axiomatize the algebraic properties of the ring $A$
needed for the proofs to go through (crucially among them, the Weierstrass
Division Property). As a consequence, for example, Theorem C also holds for
the subring $A={\mathbb{Q}}[[X]]$ of ${\mathbb{Q}}_{p}[[X]]$, the ring
$A={\mathbb{Q}}_{p}[[X]]^{\operatorname{a}}$ of power series in
${\mathbb{Q}}_{p}[[X]]$ which are algebraic over ${\mathbb{Q}}_{p}(X)$, or the
ring $A={\mathbb{Q}}_{p}[[X]]^{\operatorname{da}}$ of differentially algebraic
power series. (These facts can also be deduced from the Artin Approximation
Theorem as proved in [17], but this would be overkill.)

### Restricted analytic variants

Both Theorems A and B above deal with germs of $p$-adic analytic functions at
$0$. The subring ${\mathbb{Q}}_{p}\langle X\rangle$ of
${\mathbb{Q}}_{p}\\{X\\}$ consisting of those power series which converge for
all $a\in{\mathbb{Q}}_{p}^{m}$ with $\lvert a\rvert_{p}\leqslant 1$ also
satisfies a Weierstrass Division Theorem (albeit of a slightly different kind
than in ${\mathbb{Q}}_{p}\\{X\\}$). Our method is adaptable to this setting,
and so we also obtain versions of Theorems A and B for the ring
$A={\mathbb{Q}}_{p}\langle X\rangle$ of restricted (sometimes called strictly
convergent) power series over ${\mathbb{Q}}_{p}$, where the condition “for
each $a\in{\mathbb{Q}}_{p}^{m}$ sufficiently close to $0$” in both cases is
replaced by “for each $a\in{\mathbb{Q}}_{p}^{m}$ with ${\lvert
a\rvert_{p}\leqslant 1}$”. These theorems also hold for certain subrings of
${\mathbb{Q}}_{p}\langle X\rangle$ such as the ring ${\mathbb{Q}}_{p}\langle
X\rangle^{\operatorname{a}}:={\mathbb{Q}}_{p}[[X]]^{\operatorname{a}}\cap{\mathbb{Q}}_{p}\langle
X\rangle$. (See Corollaries 7.35 and 7.36 for the precise statements.)

The same methods also yield a restricted analytic Nullstellensatz and
Hilbert’s 17th Problem for real closed fields ${{\boldsymbol{k}}}$ equipped
with a complete ultrametric absolute value $\lvert\,\cdot\,\rvert$, such as
the Levi-Civita field: the completion of the group algebra
${\mathbb{R}}[t^{{\mathbb{Q}}}]$ equipped with the ultrametric absolute value
satisfying $\lvert at^{q}\rvert=\operatorname{e}^{-q}$ for
$a\in{\mathbb{R}}^{\times}$, ${q\in{\mathbb{Q}}}$. (Cf. [40, Chapter 1, §7].)
Let ${{\boldsymbol{k}}}\langle X\rangle$ denote the subring of
${{\boldsymbol{k}}}[[X]]$ consisting of those power series which converge for
all $a\in{{\boldsymbol{k}}}^{m}$ with $\lvert a\rvert\leqslant 1$.

######  Theorem D.

Let $f,g_{1},\dots,g_{n}\in{{\boldsymbol{k}}}\langle X\rangle$. Then

  1. (1)

$g_{1}(a)=\cdots=g_{n}(a)=0\Rightarrow f(a)=0$ for all
$a\in{{\boldsymbol{k}}}^{m}$ with $\lvert a\rvert\leqslant 1$ if and only if
there are $k\geqslant 1$ and
$b_{1},\dots,b_{l},h_{1},\dots,h_{n}\in{{\boldsymbol{k}}}\langle X\rangle$
such that $f^{2k}+b_{1}^{2}+\cdots+b_{l}^{2}=g_{1}h_{1}+\cdots+g_{n}h_{n}$;
and

  2. (2)

$f(a)\geqslant 0$ for all $a\in{{\boldsymbol{k}}}^{m}$ with $\lvert
a\rvert\leqslant 1$ if and only if there are non-zero
$g,h_{1},\dots,h_{k}\in{{\boldsymbol{k}}}\langle X\rangle$ such that
$g^{2}f=h_{1}^{2}+\cdots+h_{k}^{2}$.

(Similarly with ${{\boldsymbol{k}}}\langle
X\rangle^{\operatorname{a}}:={{\boldsymbol{k}}}[[X]]^{\operatorname{a}}\cap{{\boldsymbol{k}}}\langle
X\rangle$ in place of ${{\boldsymbol{k}}}\langle X\rangle$.)

A version of part (2) of this theorem for ${{\boldsymbol{k}}}[X]$ in place of
${{\boldsymbol{k}}}\langle X\rangle$ was shown by Dickmann [18, Theorem 2].
For our final theorem (an analogue of Theorem B for ${{\boldsymbol{k}}}\langle
X\rangle$) we let $H_{K}$ be the subring of
$K:=\operatorname{Frac}({{\boldsymbol{k}}}\langle X\rangle)$ generated by the
elements $1/(1+h)$ where $h$ is sum of squares in $K$; this “real” analogue of
the $p$-adic Kochen ring $\Lambda_{F}$ from above was introduced by Schülting
[57]. We also let $R:=\\{a\in{{\boldsymbol{k}}}:\lvert a\rvert\leqslant 1\\}$
be the valuation ring of ${{\boldsymbol{k}}}$, with maximal ideal
$\mathfrak{m}:=\\{a\in{{\boldsymbol{k}}}:\lvert a\rvert<1\\}$, and we let
$R\langle X\rangle$ be the subring of ${{\boldsymbol{k}}}\langle X\rangle$
consisting of all $f\in{{\boldsymbol{k}}}\langle X\rangle$ with $\lvert
f(a)\rvert\leqslant 1$ for all $a\in R^{m}$. The ring $H_{K}$ is a Prüfer
domain; hence so is the subring $H_{K}R\langle X\rangle$ of $K$ generated by
$H_{K}$ and $R\langle X\rangle$. (In particular, $H_{K}R\langle X\rangle$ is
integrally closed.)

######  Theorem E.

Let $f,g\in{{\boldsymbol{k}}}\langle X\rangle$. Then

$\lvert f(a)\rvert\leqslant\lvert g(a)\rvert$ for all $a\in R^{m}$
$\displaystyle\quad\Longleftrightarrow\quad f\in g\,H_{K}R\langle X\rangle,$
$\lvert f(a)\rvert<\lvert g(a)\rvert$ for all $a\in R^{m}$
$\displaystyle\quad\Longleftrightarrow\quad f\in g\,\mathfrak{m}H_{K}R\langle
X\rangle.$

The theorems above permit various further potential generalizations which we
have not attempted here. For example, let $E$ be a finite extension of
${\mathbb{Q}}_{p}$. Jarden-Roquette [30] proved a Nullstellensatz for ideals
in $E[X]$ and established the $p$-adic version of Hilbert’s 17th Problem for
rational functions in $E(X)$. (Here, the definition of the $p$-adic Kochen
operator needs to be suitably modified.) It should be routine to adapt
Theorems A, B, C to this setting. Less routine would be an adaptation of our
theorems to germs of analytic functions on basic $p$-adic subsets of analytic
varieties, analogous to [30, Theorems 1.2, 7.2]. (See also [48, §7].) It would
also be interesting to generalize Theorems D and E in a similar way. (For
polynomials this was studied in [29, 39].) It would also be interesting to use
Theorem C in a similar way as Merrien used his Nullstellensatz for formal
power series over the reals in order to obtain a $p$-adic analogue of his
Nullstellensatz for germs of $C^{\infty}$-functions; see [24, Conjecture 4.3].
(Here one should employ the concept of strict differentiability for
multivariate $p$-adic functions developed in [7, 25, 58], in order to avoid
the well-known disadvantages of the usual notion of differentiability over the
$p$-adics caused by the failure of the Mean Value Theorem.)

### Organization of the paper

After a first preliminary section, in Section 2 we recall basic facts about
Weierstrass systems and establish the setting of infinitesimal Weierstrass
structures, which provide a convenient framework to state theorems like the
ones discussed earlier in this introduction. As a warm-up, since these cases
are conceptually easier than the $p$-adic version, in Sections 3 and 4 we then
revisit Rückert’s and Risler’s Nullstellensätze using our method. Here, the
relevant model-theoretic results are quantifier elimination theorems for the
theory $\operatorname{ACVF}$ of algebraically closed non-trivially valued
fields (A. Robinson) and for the theory $\operatorname{RCVF}$ of real closed
non-trivially convexly valued fields (Cherlin-Dickmann), respectively. The
model-theoretic ingredient for the proofs of Theorems A, B, C is an analogous
elimination theorem for the theory $p\operatorname{CVF}$ of $p$-adically
closed non-trivially $p$-convexly valued fields. We discuss this theory and
the relevant QE theorem (which is less prominent in the literature than those
for $\operatorname{ACVF}$ and $\operatorname{RCVF}$: but see [6, 28]) in
Section 5, before giving the proofs of our theorems in Section 6. Finally,
Section 7 contains the restricted analytic adaptations of these results,
including proofs of Theorems D and E.

### Related work

The Weierstrass systems used here are close to those of [17, 20]. Cluckers-
Lipshitz [13] introduced a very general class of subrings of formal power
series rings stable under Weierstrass Division, called “separated Weierstrass
systems”, which in a sense amalgamates the Weierstrass systems used here and
our restricted Weierstrass systems from Definition 7.8. Given a separated
Weierstrass system $\mathcal{A}$, this leads to the concept of a “separated
analytic $\mathcal{A}$-structure” on a henselian valued field, expanding our
Definition 2.12, and in the case of characteristic zero, to a quantifier
elimination in a certain multi-sorted language [13, Theorem 6.3.7]. In the
algebraically closed case (without assumptions on the characteristic) one also
has quantifier elimination in a one-sorted language [13, Theorem 4.5.15]. A
one-sorted QE for the valued field ${\mathbb{Q}}_{p}$ expanded by primitives
for the functions defined by the restricted power series was proved by Denef
and van den Dries [16]. Cubides Kovacics and Haskell [14] similarly show that
the real closed ordered valued field ${{\boldsymbol{k}}}$ from the context of
Theorems D and E with a separated analytic $\mathcal{A}$-structure where
$\mathcal{A}$ contains the rings $R\langle X\rangle$ of restricted power
series, admits quantifier elimination in a natural one-sorted language.
Bleybel [8] proves a quantifier elimination theorem for the $p$-adic analogue
of the Levi-Civita field (i.e., the completion of the group algebra
${\mathbb{Q}}_{p}[t^{\mathbb{Q}}]$ equipped with the ultrametric absolute
value satisfying $\lvert at^{q}\rvert=\operatorname{e}^{-q}$ for
$a\in{\mathbb{Q}}_{p}^{\times}$, $q\in{\mathbb{Q}}$) in a $3$-sorted language
which contains function symbols for (among other things) the functions on the
valuation ring given by overconvergent $p$-adic power series. We do not know
whether these QE theorems for valued fields with analytic structure can be
used to give alternative proofs of our Theorems A–E.

### Notational conventions

Throughout this paper $k$, $l$, $m$, $n$ range over the set
${\mathbb{N}}=\\{0,1,2,\dots\\}$ of natural numbers, and $p$ is a prime
number. In this paper all rings are commutative with $1$. Let $R$ be a ring.
We let $R^{\times}$ be the group of units of $R$, and if $R$ is an integral
domain, then we denote the fraction field of $R$ by $\operatorname{Frac}(R)$.

##  1\. Valuation-Theoretic Preliminaries

In this paper a valued field is a field equipped with one of its valuation
rings. Let $K$ be a valued field. Unless specified otherwise we let
$\mathcal{O}$ be the distinguished valuation ring of $K$, with maximal ideal
$\smallo$. The residue field of $K$ is
$\operatorname{res}(K):=\mathcal{O}/\smallo$, with residue morphism
$a\mapsto\overline{a}=a+\smallo\colon\mathcal{O}\to\operatorname{res}(K)$. The
value group of $K$ is the ordered abelian group
$\Gamma=K^{\times}/\mathcal{O}^{\times}$, written additively, with ordering
$a\mathcal{O}^{\times}\leqslant b\mathcal{O}^{\times}$ iff
$b/a\in\mathcal{O}$. The map $a\mapsto va=a\mathcal{O}^{\times}\colon
K^{\times}\to\Gamma$ is a valuation on $K$. If we need to indicate the
dependence on $K$ we attach a subscript $K$, so $\mathcal{O}=\mathcal{O}_{K}$,
$v=v_{K}$, etc. If $K\subseteq L$ is an extension of valued fields (that is,
$K\subseteq L$ is a field extension and $\mathcal{O}=\mathcal{O}_{L}\cap K$),
then we identify $\operatorname{res}(K)$ with a subfield of
$\operatorname{res}(L)$ and $\Gamma$ with an ordered subgroup of $\Gamma_{L}$
in the natural way. For $a,b\in K$ we set

$a\preceq b\,:\Leftrightarrow\,va\geqslant vb,\quad a\prec
b\,:\Leftrightarrow\,va>vb,\quad a\asymp b\,:\Leftrightarrow\,va=vb,\quad
a\sim b\,:\Leftrightarrow\,a-b\prec a.$

The binary relation $\preceq$ is an example of a dominance relation on $K$:

######  Definition 1.1.

A dominance relation on an integral domain $R$ is a binary relation $\preceq$
on $R$ such that for all $f,g,h\in R$:

  * (D1)

$1\not\preceq 0$;

  * (D2)

$f\preceq f$;

  * (D3)

$f\preceq g\text{ and }g\preceq h\Rightarrow f\preceq h$;

  * (D4)

$f\preceq g$ or $g\preceq f$;

  * (D5)

$f\preceq g\Leftrightarrow fh\preceq gh$, provided $h\neq 0$;

  * (D6)

$f\preceq h\text{ and }g\preceq h\Rightarrow f+g\preceq h$.

Thus, if $\mathcal{O}$ is a valuation ring of $K$, we obtain a dominance
relation on $K$ via

$f\preceq g\ :\Longleftrightarrow\ vf\geqslant vg\ \Longleftrightarrow\
\text{$f=gh$ for some $h\in\mathcal{O}$.}$ (1.1)

Conversely, if $\preceq$ is a dominance relation on $K$, then clearly
$\mathcal{O}:=\\{f\in K:f\preceq 1\\}$ is a valuation ring of $K$, and if $v$
denotes the corresponding valuation on $K$, then the equivalence (1.1) holds,
for all $f,g\in K$. We call $\mathcal{O}$ the valuation ring associated to the
dominance relation $\preceq$. This yields a one-to-one correspondence between
dominance relations on $K$ and valuation rings of $K$.

Let $R$ be an integral domain. The trivial dominance relation
$\preceq_{\operatorname{t}}$ on $R$ is the one with
$r\preceq_{\operatorname{t}}s$ for all $r,s\in R$ with $s\neq 0$. For any
dominance relation $\preceq$ on $R$ there is a unique dominance relation
$\preceq_{F}$ on $F=\operatorname{Frac}(R)$ such that
$(R,{\preceq})\subseteq(F,{\preceq_{F}})$:

$\frac{r_{1}}{s}\preceq_{F}\frac{r_{2}}{s}\quad\Longleftrightarrow\quad
r_{1}\preceq r_{2}\qquad(r_{1},r_{2},s\in R,\ s\neq 0).$

######  Example.

The non-trivial dominance relations on ${\mathbb{Z}}$ are exactly the $p$-adic
dominance relations $\preceq_{p}$, given by $a\preceq_{p}b$ iff for all $n$:
$b\in p^{n}{\mathbb{Z}}\Rightarrow a\in p^{n}{\mathbb{Z}}$.

###  1.1. Model-theoretic treatment of valued fields

We now briefly explain basic model-theoretic terminology and notation,
focussing on valued fields. A concise presentation of all the general model
theory needed here is in [3, Appendix B]. A textbook which also pays attention
to valued fields is [46]; for more specialized expositions of the model theory
of valued fields see [11, 21].

A (one-sorted) language $\mathcal{L}$ is a pair
$(\mathcal{L}^{\operatorname{r}},\mathcal{L}^{\operatorname{f}})$ where
$\mathcal{L}^{\operatorname{r}}$, $\mathcal{L}^{\operatorname{f}}$ are
disjoint (possibly empty, or infinite) collections of relation symbols and
function symbols, respectively. Each of these symbols has an associated arity
(a natural number); symbols of arity $n$ are also said to be $n$-ary. A
$0$-ary function symbol is called a constant symbol. It is customary to
present a language as a disjoint union
$\mathcal{L}=\mathcal{L}^{\operatorname{r}}\cup\mathcal{L}^{\operatorname{f}}$
of its sets of relation and function symbols, while separately specifying the
arities of the various symbols. For example,
$\mathcal{L}_{\operatorname{R}}=\\{0,1,{-},{+},{\,\cdot\,}\\}$ is the language
of rings: here $0$, $1$ are constant symbols, $-$ is a unary function symbol,
and $+$, $\cdot$ are binary function symbols (so
$\mathcal{L}_{\operatorname{R}}$ has no relation symbols). Another example,
relevant for this paper, is the expansion
$\mathcal{L}_{\preceq}=\mathcal{L}_{\operatorname{R}}\cup\\{\preceq\\}$ of
$\mathcal{L}_{\operatorname{R}}$ by a binary relation symbol $\preceq$. Here,
$\mathcal{L}$ is an expansion of a language $\mathcal{L}_{0}$, and
$\mathcal{L}_{0}$ a reduct of $\mathcal{L}$, if
$\mathcal{L}_{0}^{\operatorname{r}}\subseteq\mathcal{L}^{\operatorname{r}}$
and
$\mathcal{L}_{0}^{\operatorname{f}}\subseteq\mathcal{L}^{\operatorname{f}}$,
with unchanged arities for the symbols in $\mathcal{L}_{0}$.

Let $\mathcal{L}$ be a language. An $\mathcal{L}$-structure $\mathbf{M}$
consists of its underlying set $M\neq\emptyset$ together with interpretations
of each symbol of $\mathcal{L}$ in $\mathbf{M}$: for each $m$-ary
$R\in\mathcal{L}^{\operatorname{r}}$, a relation $R^{\mathbf{M}}\subseteq
M^{m}$, and for $n$-ary $f\in\mathcal{L}^{\operatorname{f}}$, a function
$f^{\mathbf{M}}\colon M^{n}\to M$. (So each constant symbol $c$ is interpreted
by a function $M^{0}\to M$, identified with the unique element
$c^{\mathbf{M}}$ in its image, $M^{0}$ being a singleton.) In practice the
superscript $\mathbf{M}$ is often suppressed, thus a structure and its
universe are denoted by the same letter, and so are the symbols of
$\mathcal{L}$ and their interpretations in $\mathbf{M}$. Morphisms,
embeddings, and substructures of $\mathcal{L}$-structures are defined in the
natural way; e.g., an $\mathcal{L}$-structure $\mathbf{M}$ is a substructure
of an $\mathcal{L}$-structure $\mathbf{N}$ if $M\subseteq N$ and
$R^{\mathbf{M}}=R^{\mathbf{N}}\cap M^{m}$ and
$f^{\mathbf{N}}|_{M^{n}}=f^{\mathbf{M}}$ for each $m$-ary
$R\in\mathcal{L}^{\operatorname{r}}$ and $n$-ary
$f\in\mathcal{L}^{\operatorname{f}}$. If $\mathcal{L}_{0}$ is a reduct of
$\mathcal{L}$, then $\mathbf{M}|\mathcal{L}_{0}$ is the
$\mathcal{L}_{0}$-reduct of $\mathbf{M}$: the $\mathcal{L}_{0}$-structure with
same underlying set $M$ and the same interpretations of the symbols in
$\mathcal{L}_{0}$ as in $\mathbf{M}$.

Each ring $R$ can be construed as an
$\mathcal{L}_{\operatorname{R}}$-structure by interpreting $0$, $1$ by the
additive and multiplicative identity of $R$, respectively, $-$ by the function
$r\mapsto-r$, and $+$, $\cdot$ by $(r,s)\mapsto r+s$ and $(r,s)\mapsto r\cdot
s$, respectively. Morphisms, embeddings, and substructures of
$\mathcal{L}_{\operatorname{R}}$-structures then correspond to morphisms,
embeddings, and substructures of rings, respectively. (But of course we are,
in principle, free to also view $R$ as an
$\mathcal{L}_{\operatorname{R}}$-structure in an unnatural way, by choosing
the interpretations of the symbols of $\mathcal{L}_{\operatorname{R}}$ any way
we like.) In this paper we also always construe each valued field
$(K,\mathcal{O})$ as an $\mathcal{L}_{\preceq}$-structure by interpreting
$\preceq$ as the dominance relation on $K$ given by (1.1). Then “valued
subfield” corresponds to “$\mathcal{L}_{\preceq}$-substructure.” (This way of
treating valued fields as model theoretic structures is essentially equivalent
to that in [46, Chapter 4], which employs valuation divisibilities instead of
dominance relations. For other choices see [11, 21].)

The purpose of a language $\mathcal{L}$ is to generate $\mathcal{L}$-formulas,
which express properties of (finite tuples of) elements of
$\mathcal{L}$-structures. They are (well-formed, in a certain precise sense)
finite words on an alphabet consisting of
$\mathcal{L}^{\operatorname{r}}\cup\mathcal{L}^{\operatorname{f}}$ together
with the symbols $\top$, $\perp$, $\neg$, $\vee$, $\wedge$, $=$, $\forall$,
$\exists$, to be thought of as true, false, not, or, and, equals, there
exists, and for all, respectively, together with an infinite collection of
variables (usually written $x$, $y$, $z$, …), thought of as ranging over the
universe of an $\mathcal{L}$-structure (and not over, say, subsets thereof).
Variables which do not appear within the scope of a quantifier in an
$\mathcal{L}$-formula $\varphi$ are said to be free in $\varphi$, and
$\mathcal{L}$-formulas without free variables are $\mathcal{L}$-sentences. For
example, the commutativity of addition in a ring is expressed by the
$\mathcal{L}_{\operatorname{R}}$-sentence $\forall x\forall y(x+y=y+x)$.
Strictly speaking, this should be $\forall x\forall y=+xy+yx$; but as usual we
take the liberty of writing binary symbols, including $=$, in infix rather
than prefix notation, and of freely employing parentheses to enhance
readability. Below $\varphi$, $\psi$, $\theta$ range over
$\mathcal{L}$-formulas and $\sigma$ over $\mathcal{L}$-sentences. We also use
common abbreviations like $\forall x$ for $\forall x_{1}\cdots\forall x_{m}$,
where $x=(x_{1},\dots,x_{m})$ is a tuple of distinct variables, and
$\varphi\leftrightarrow\psi$ in place of
$(\varphi\wedge\psi)\vee(\neg\varphi\wedge\neg\psi)$.

Given $x$ as above, writing $\varphi(x)$ indicates that $x$ contains all free
variables in $\varphi$. For an $\mathcal{L}$-structure $\mathbf{M}$ and $a\in
M^{m}$ one then defines, by induction on the construction of $\varphi$, when
$\mathbf{M}$ satisfies $\varphi(a)$—in symbols:
$\mathbf{M}\models\varphi(a)$—in the natural way. The subset of $M^{m}$
defined by $\varphi(x)$ in $\mathbf{M}$ is
$\varphi^{\mathbf{M}}:=\big{\\{}a\in
M^{m}:\mathbf{M}\models\varphi(a)\big{\\}}$. E.g., for a valued field $K$ and
the $\mathcal{L}_{\preceq}$-formulas $\varphi(x)$, $\psi(x)$, $\theta(x)$,
where $x$ is a single variable, given by $x\preceq 1$, ${x\preceq
1}\wedge{\neg 1\preceq x}$, and ${x\preceq 1}\wedge\exists y({x\cdot
y=1}\wedge y\preceq 1)$, respectively, we have $\mathcal{O}=\varphi^{K}$,
$\smallo=\psi^{K}$, and $\mathcal{O}^{\times}=\theta^{K}$. Note: if $y$ is a
variable which neither occurs in $x$ nor free in $\varphi$, then for $a\in
M^{m}$, $b\in M$ we have $\mathbf{M}\models\varphi(a,b)$ iff
$\mathbf{M}\models\varphi(a)$; in particular, the status of
$\mathbf{M}\models\sigma$ does not depend on $a$. The $\mathcal{L}$-structures
$\mathbf{M}$, $\mathbf{N}$ are said to be elementarily equivalent
($\mathbf{M}\equiv\mathbf{N}$) if for each $\sigma$, $\mathbf{M}\models\sigma$
iff $\mathbf{N}\models\sigma$.

Let $\Sigma$ be a set of $\mathcal{L}$-sentences. Then $\mathbf{M}$ is a model
of $\Sigma$ (in symbols: $\mathbf{M}\models\Sigma$) if
$\mathbf{M}\models\sigma$ for all $\sigma\in\Sigma$. We also write
$\Sigma\models\sigma$ if every model of $\Sigma$ satisfies $\sigma$. If
$\Sigma$ contains each $\sigma$ with $\Sigma\models\sigma$, then $\Sigma$ is
an $\mathcal{L}$-theory. Clearly
$\Sigma^{\models}:={\\{\sigma:\Sigma\models\sigma\\}}$ is the smallest
$\mathcal{L}$-theory containing $\Sigma$. Taking
$\mathcal{L}=\mathcal{L}_{\preceq}$ and for $\Sigma$ the set of
$\mathcal{L}_{\preceq}$-sentences which consists of the axioms for fields
(which can be formulated in the sublanguage $\mathcal{L}_{\operatorname{R}}$
of $\mathcal{L}_{\preceq}$) together with (universal)
$\mathcal{L}_{\preceq}$-sentences formalizing the properties (D1)–(D6), we
obtain the $\mathcal{L}_{\preceq}$-theory
$\operatorname{VF}:=\Sigma^{\models}$ of valued fields. If $\Sigma$ has a
model, then $\mathbf{M}\equiv\mathbf{N}$ for all
$\mathbf{M},\mathbf{N}\models\Sigma$ iff for all $\sigma$, either
$\Sigma\models\sigma$ or $\Sigma\models\neg\sigma$; in this case, $\Sigma$ is
said to be complete. The $\mathcal{L}_{\preceq}$-theory $\operatorname{VF}$ is
not complete since (for example) it has models of different characteristic.

If $A\subseteq M$, we let $\mathcal{L}_{A}$ be the expansion of $\mathcal{L}$
by a new constant symbol $\underline{a}$ for each $a\in A$, and
$\mathbf{M}_{A}$ be the $\mathcal{L}_{A}$-structure with $\mathcal{L}$-reduct
$\mathbf{M}$ where each $\underline{a}$ is interpreted by $a$. We say that
$\mathbf{M}$ is an elementary substructure of $\mathbf{N}$, and $\mathbf{N}$
an elementary extension of $\mathbf{M}$, if $M\subseteq N$ and
$\mathbf{M}_{M}\equiv\mathbf{N}_{M}$. If every extension of models of $\Sigma$
is elementary, then $\Sigma$ is said to be model complete. Finally, we say
that $\varphi$ is quantifier-free if $\forall$, $\exists$ do not occur in
$\varphi$, and $\Sigma$ is said to admit quantifier elimination if for each
$\varphi(x)$, where $x=(x_{1},\dots,x_{m})$, there is a quantifier-free
$\psi(x)$ with $\Sigma\models\forall x(\varphi\leftrightarrow\psi)$. In this
case, $\Sigma$ is also model complete.

###  1.2. Extension of valuations

Let $A$ be a local subring of a field $K$ with maximal ideal
$\mathfrak{m}=\mathfrak{m}_{A}$ and residue field
${{\boldsymbol{k}}}=A/\mathfrak{m}$. We recall that any maximal element of the
class of all local subrings of $K$ lying over $A$, partially ordered by
$B\leqslant B^{\prime}:\Longleftrightarrow$ $B^{\prime}$ lies over $B$, is a
valuation ring of $K$. (Krull; cf., e.g., [3, Proposition 3.1.13].) In
particular, by Zorn, there is always a valuation ring of $K$ lying over $A$.
Under suitable hypotheses on $A$ we can be more precise (see, e.g., [30, Lemma
A1]):

######  Lemma 1.2.

Suppose $A$ is regular and $K=\operatorname{Frac}(A)$. Then there is a
valuation ring $\mathcal{O}$ of $K$ lying over $A$ such that the natural
inclusion $A\to\mathcal{O}$ induces an isomorphism
${{\boldsymbol{k}}}\to\mathcal{O}/\smallo$.

(Indeed, for each regular system of parameters $x_{1},\dots,x_{d}$ of $A$
there is a unique valuation ring $\mathcal{O}$ as in Lemma 1.2 with
$\Gamma={\mathbb{Z}}v(x_{1})\oplus\cdots\oplus{\mathbb{Z}}v(x_{d})$, ordered
lexicographically.)

###  1.3. Coarsening and specialization

Let $K$ be a valued field and $\Delta$ be a convex subgroup of
$\Gamma=v(K^{\times})$. We have the ordered quotient group
$\dot{\Gamma}=\Gamma/\Delta$ and the valuation

$\dot{v}=v_{\Delta}\colon
K^{\times}\to\dot{\Gamma},\qquad\dot{v}(a):=v(a)+\Delta\text{ for $a\in
K^{\times}$}$

on the field $K$, the $\Delta$-coarsening of $v$. The valuation ring of
$\dot{v}$ is

$\dot{\mathcal{O}}\ =\ \mathcal{O}_{\Delta}\ :=\ \\{a\in
K:\text{$va\geqslant\delta$ for some $\delta\in\Delta$}\\},$

which has $\mathcal{O}$ as a subring and has maximal ideal

$\dot{\smallo}\ :=\ \\{a\in K:va>\Delta\\}\subseteq\smallo.$

The valued field $(K,\dot{\mathcal{O}})$ is called the $\Delta$-coarsening of
$K$. Put $\dot{K}:=\dot{\mathcal{O}}/\dot{\smallo}$, the residue field of
$\dot{\mathcal{O}}$, and $\dot{a}:=a+\dot{\smallo}\in\dot{K}$ for
$a\in\dot{\mathcal{O}}$. Then for
$a\in\dot{\mathcal{O}}\setminus\dot{\smallo}$ the value $va$ depends only on
the residue class $\dot{a}\in\dot{K}$. This gives the valuation
$v\colon\dot{K}^{\times}\to\Delta$, $v\dot{a}:=va$ with valuation ring
$\mathcal{O}_{\dot{K}}=\\{\dot{a}:a\in\mathcal{O}\\}$, having maximal ideal
$\smallo_{\dot{K}}=\\{\dot{a}:a\in\smallo\\}$. Throughout $\dot{K}$ stands for
the valued field $(\dot{K},\mathcal{O}_{\dot{K}})$, called the
$\Delta$-specialization of $K$. The composed map

$\mathcal{O}\to\mathcal{O}_{\dot{K}}\to\operatorname{res}(\dot{K})=\mathcal{O}_{\dot{K}}/\smallo_{\dot{K}}$

has kernel $\smallo$, and thus induces a field isomorphism
$\operatorname{res}(K)\overset{\cong}{\longrightarrow}\operatorname{res}(\dot{K})$,
and we identify $\operatorname{res}(K)$ and $\operatorname{res}(\dot{K})$ via
this map.

Every subring of $K$ containing $\mathcal{O}$ is a valuation ring of $K$. In
fact, every such subring $\mathcal{O}_{1}\supseteq\mathcal{O}$ of $K$ arises
as the valuation ring of the $\Delta_{1}$-coarsening of $v$ for some convex
subgroup $\Delta_{1}$ of $\Gamma$, namely
$\Delta_{1}:=v(\mathcal{O}_{1}\setminus\\{0\\})$. The map
$\mathcal{O}_{1}\mapsto\Delta_{1}$ is an inclusion-preserving bijection from
the set of subrings of $K$ containing $\mathcal{O}$ onto the set of convex
subgroups of $\Gamma$. (Cf., e.g., [3, Lemmas 3.1.4, 3.1.5].) In particular,
the collection of subrings of $K$ containing $\mathcal{O}$ is totally ordered
under inclusion, and we have $\mathcal{O}_{1}\supset\mathcal{O}$ iff
$\Delta_{1}\neq\\{0\\}$. The following is easy to verify:

######  Lemma 1.3.

Let ${{\boldsymbol{k}}}:=\operatorname{res}(K)$ and
$\mathcal{O}_{{{\boldsymbol{k}}}}$ be a valuation ring of
${{\boldsymbol{k}}}$. Then
$\mathcal{O}_{0}:={\\{a\in\mathcal{O}:\overline{a}\in\mathcal{O}_{{{\boldsymbol{k}}}}\\}}$
is a valuation ring of $K$. If $\Delta_{0}$ is the convex subgroup of the
value group of $\mathcal{O}_{0}$ with $\dot{\mathcal{O}_{0}}=\mathcal{O}$,
then the $\Delta_{0}$-specialization of $(K,\mathcal{O}_{0})$ is
$({{\boldsymbol{k}}},\mathcal{O}_{{{\boldsymbol{k}}}})$.

###  1.4. Convexity

Let $K$ be an ordered field. Recall that a subring of $K$ is convex iff it
contains the interval $[0,1]$, and as a consequence, every convex subring of
$K$ is a valuation ring of $K$. (See, e.g., [3, Lemma 3.5.10].) Let $R$ be a
subring of $K$; then the convex hull

$A:=\big{\\{}a\in K:\text{$\lvert a\rvert\leqslant\varepsilon$ for some
$\varepsilon\in R^{>}$}\big{\\}}$

of $R$ in $K$ is a subring of $K$. This is the smallest convex subring of $K$
containing $R$, with maximal ideal

$\mathfrak{m}=\big{\\{}a\in K:\text{$\lvert a\rvert\leqslant 1/\varepsilon$
for all $\varepsilon\in R^{>}$}\big{\\}}.$

Let $\mathcal{O}_{0}$ be the convex hull of ${\mathbb{Q}}$ in $K$. Then a
subring of $K$ is convex iff it contains $\mathcal{O}_{0}$; in particular, the
collection of convex subrings of $K$ is totally ordered under inclusion. We
generalize these concepts to an arbitrary valued field. Thus let now $K$ be a
valued field with valuation ring $\mathcal{O}$ and associated dominance
relation $\preceq$.

######  Definition 1.4.

We say that a subring of $K$ is convex (in the valued field $K$) if it
contains $\mathcal{O}$. Clearly $\mathcal{O}$ and $K$ are convex subrings of
$K$; we call these the trivial convex subrings of $K$.

Note that every convex subring of $K$ is a valuation ring of $K$. In fact, for
each convex subgroup $\Delta$ of the value group of $K$, the valuation ring of
the $\Delta$-coarsening of $K$ is a convex subring of $K$, and each convex
subring of $K$ arises this way. Thus the set of convex subrings of $K$ is
totally ordered under inclusion.

######  Lemma 1.5.

Let $R$ be a subring of $K$. Then the subring $A=\mathcal{O}R$ of $K$
generated by $\mathcal{O}$ and $R$ is the smallest convex subring of $K$ which
contains $R$. We have

$A=\big{\\{}a\in K:\text{$a\preceq\varepsilon$ for some $\varepsilon\in
R$}\big{\\}},$

with maximal ideal

$\mathfrak{m}=\big{\\{}a\in K:\text{$a\prec 1/\varepsilon$ for all non-zero
$\varepsilon\in R$}\big{\\}}.$

###### Proof.

The first statement is easy to verify; for the second statement use that the
maximal ideal of the local ring $A$ is given by $\mathfrak{m}=\\{a\in
A:\text{$a=0$ or $a\notin A^{\times}$}\\}$. ∎

Given a subring $R$ of $K$, we call $A=\mathcal{O}R$ the convex hull of $R$ in
$K$, denoted by $\operatorname{conv}_{K}(R)$, or by $\operatorname{conv}(R)$
if $K$ is clear from the context. (If $K$ comes equipped with a field ordering
and $\mathcal{O}=\mathcal{O}_{0}$ as above, then this agrees with the convex
hull of $R$ in the ordered field $K$.) The map
$R\mapsto\operatorname{conv}(R)$ is a closure operator on the collection of
subrings of $K$, that is,

$R\subseteq\operatorname{conv}(R),\quad R\subseteq S\ \Longrightarrow\
\operatorname{conv}(R)\subseteq\operatorname{conv}(S),\quad\operatorname{conv}(\operatorname{conv}(R))=\operatorname{conv}(R).$

Let $L$ be a valued field extension of $K$ and $R$ be a convex subring of $K$.
It is easily seen that $\operatorname{conv}_{L}(R)$ lies over $R$. Moreover
(cf. [3, remarks after Proposition 3.1.20]):

######  Lemma 1.6.

If $L$ is algebraic over $K$, then $\operatorname{conv}_{L}(R)$ is the unique
convex subring of $L$ lying over $R$.

We conclude this section with some reminders about ${\mathbb{Z}}$-groups.

###  ${\mathbb{Z}}$-groups

Let $\Gamma$ be an ordered abelian group, written additively. We let
$\Gamma^{>}:={\\{\gamma\in\Gamma:a>0\\}}$, and similarly with $\geqslant$
instead of $>$. If $\Gamma^{>}$ has smallest element, denoted by $1$, then we
identify ${\mathbb{Z}}$ with an ordered subgroup of $\Gamma$ via the embedding
$r\mapsto r\cdot 1\colon{\mathbb{Z}}\to\Gamma$, making ${\mathbb{Z}}$ the
smallest non-zero convex subgroup of $\Gamma$. We recall that $\Gamma$ is said
to be a ${\mathbb{Z}}$-group if $\Gamma^{>}$ has a smallest element $1$ and
for each $n\geqslant 1$ and $\gamma\in\Gamma$ there are some
$i\in\\{0,\dots,n-1\\}$ such that $\gamma-i\in n\Gamma$. The ordered abelian
group ${\mathbb{Z}}$ of integers is a ${\mathbb{Z}}$-group, and every
${\mathbb{Z}}$-group is elementarily equivalent to ${\mathbb{Z}}$ in the
language $\\{0,{-},{+},{\leqslant}\\}$ of ordered abelian groups. (See, e.g.,
[46, §4.1].) For future reference here are a few other well-known (and easy to
prove) facts :

######  Lemma 1.7.

Let $\Delta\neq\\{0\\}$ be a convex subgroup of $\Gamma$. Then $\Gamma$ is a
${\mathbb{Z}}$-group iff $\Delta$ is a ${\mathbb{Z}}$-group and
$\Gamma/\Delta$ is divisible.

######  Lemma 1.8.

Let $\Delta$ be a subgroup of a ${\mathbb{Z}}$-group $\Gamma$ which contains
the smallest element of $\Gamma^{>}$. Then $\Delta$ is a ${\mathbb{Z}}$-group
iff $\Gamma/\Delta$ is torsion-free.

##  2\. Weierstrass Systems and Infinitesimal Analytic Structures

In this section we let ${{\boldsymbol{k}}}$ be a field, of arbitrary
characteristic unless stated otherwise. We let $(X_{m})_{m\geqslant 1}$ be a
sequence of distinct indeterminates over ${{\boldsymbol{k}}}$. As usual
${{\boldsymbol{k}}}[[X_{1},\dots,X_{m}]]$ denotes the ring of formal power
series over ${{\boldsymbol{k}}}$ in the indeterminates $X_{1},\dots,X_{m}$. In
the following we let $X=(X_{1},\dots,X_{m})$, we let $\alpha$, $\beta$ range
over ${\mathbb{N}}^{m}$, and for $\alpha=(\alpha_{1},\dots,\alpha_{m})$ put
$\lvert\alpha\rvert:=\alpha_{1}+\cdots+\alpha_{m}$ and
$X^{\alpha}:=X_{1}^{\alpha_{1}}\cdots X_{m}^{\alpha_{m}}$. Consider an element

$f=\sum_{\alpha}f_{\alpha}X^{\alpha}\qquad(f_{\alpha}\in{{\boldsymbol{k}}})$

of ${{\boldsymbol{k}}}[[X]]$. If $f\neq 0$, then the order of $f$ is the
smallest $d$ such that $f_{\alpha}\neq 0$ for some $\alpha$ with
$\lvert\alpha\rvert=d$. Suppose now $m\geqslant 1$. One says that
$f\in{{\boldsymbol{k}}}[[X]]$ is _regular of order $d\in{\mathbb{N}}$ in
$X_{m}$_ if $f(0,X_{m})\in{{\boldsymbol{k}}}[[X_{m}]]$ is non-zero of order
$d$. Note that if $f=f_{1}\cdots f_{n}$ where
$f_{1},\dots,f_{n}\in{{\boldsymbol{k}}}[[X]]$, then $f$ is regular in $X_{m}$
of order $d$ iff $f_{i}$ is regular in $X_{m}$ of order $d_{i}$ for
$i=1,\dots,n$, with $d_{1}+\cdots+d_{n}=d$.

Let $d\in{\mathbb{N}}$, $d\geqslant 1$. Then we have a
${{\boldsymbol{k}}}$-algebra automorphism $\tau_{d}$ of
${{\boldsymbol{k}}}[[X]]$ such that $\tau_{d}(X_{i})=X_{i}+X_{m}^{d^{m-i}}$
for $i=1,\dots,m-1$ and $\tau_{d}(X_{m})=X_{m}$. (See, e.g., [26, I, §4, Satz
3 and the remark following it].) For non-zero $f\in{{\boldsymbol{k}}}[[X]]$
there is always some $d\in{\mathbb{N}}$, $d\geqslant 1$ such that
$\tau_{d}(f)$ is regular in $X_{m}$ (of some order). In particular, given non-
zero $f_{1},\dots,f_{n}\in{{\boldsymbol{k}}}[[X]]$, there is some
$d\in{\mathbb{N}}$, $d\geqslant 1$ such that
$\tau_{d}(f_{1}),\dots,\tau_{d}(f_{n})$ are all regular in $X_{m}$.

We also recall that ${{\boldsymbol{k}}}[[X]]$ is a complete local noetherian
integral domain with maximal ideal generated by $X_{1},\dots,X_{m}$. Thus for
$f\in{{\boldsymbol{k}}}[[X]]$ we have $f\in{{\boldsymbol{k}}}[[X]]^{\times}$
iff $f(0)\in{{\boldsymbol{k}}}^{\times}$; we say that $f$ is a $1$-unit if
$f(0)=1$.

###  2.1. Weierstrass systems

The following definition is a special instance of a concept introduced by
Denef-Lipshitz [17]:

######  Definition 2.1.

A Weierstrass system over ${{\boldsymbol{k}}}$ is a family of rings
$(W_{m})_{m\geqslant 0}$ such that for all $m$ we have, with
$X=(X_{1},\dots,X_{m})$:

  * (W1)

${{\boldsymbol{k}}}[X]\subseteq W_{m}\subseteq{{\boldsymbol{k}}}[[X]]$;

  * (W2)

for each permutation $\sigma$ of $\\{1,\dots,m\\}$, the
${{\boldsymbol{k}}}$-algebra automorphism $f(X)\mapsto
f(X_{\sigma(1)},\dots,X_{\sigma(m)})$ of ${{\boldsymbol{k}}}[[X]]$ maps
$W_{m}$ onto itself;

  * (W3)

$W_{m}\cap{{\boldsymbol{k}}}[[X^{\prime}]]=W_{m-1}$ for
$X^{\prime}=(X_{1},\dots,X_{m-1})$, $m\geqslant 1$;

  * (W4)

$W_{m}\cap{{\boldsymbol{k}}}[[X]]^{\times}=W_{m}^{\times}$; and

  * (W5)

if $g\in W_{m}$ is regular of order $d$ in $X_{m}$ ($m\geqslant 1$), then for
every $f\in W_{m}$ there are $q\in W_{m}$ and a polynomial $r\in
W_{m-1}[X_{m}]$ of degree $<d$ (in $X_{m}$) such that $f=qg+r$. (Weierstrass
Division.)

(In the case where ${{\boldsymbol{k}}}$ has positive characteristic, [17]
includes further axioms, but those are not needed for our purposes.)

Examples of Weierstrass systems include

  1. (1)

the system $\bigl{(}{{\boldsymbol{k}}}[[X_{1},\dots,X_{m}]]\bigr{)}$
consisting of all formal power series rings;

  2. (2)

the system
$\bigl{(}{{\boldsymbol{k}}}[[X_{1},\dots,X_{m}]]^{\operatorname{a}}\bigr{)}$,
where ${{\boldsymbol{k}}}[[X_{1},\dots,X_{m}]]^{\operatorname{a}}$ is the ring
of power series in ${{\boldsymbol{k}}}[[X_{1},\dots,X_{m}]]$ which are
algebraic over ${{\boldsymbol{k}}}[X_{1},\dots,X_{m}]$ (cf. [36]);

  3. (3)

if $\operatorname{char}{{\boldsymbol{k}}}=0$, the system
$\bigl{(}{{\boldsymbol{k}}}[[X_{1},\dots,X_{m}]]^{\operatorname{da}}\bigr{)}$,
where ${{\boldsymbol{k}}}[[X_{1},\dots,X_{m}]]^{\operatorname{da}}$ is the
ring of all power series
$f\in{{\boldsymbol{k}}}[[X]]={{\boldsymbol{k}}}[[X_{1},\dots,X_{m}]]$ which
are _differentially algebraic_ over ${{\boldsymbol{k}}}$, that is, the
fraction field of the subring
${{\boldsymbol{k}}}\big{[}\partial^{\lvert\beta\rvert}f/\partial
X^{\beta}\big{]}$ of ${{\boldsymbol{k}}}[[X]]$ generated by the partial
derivatives of $f$ has finite transcendence degree over ${{\boldsymbol{k}}}$
(see [20, §5]);

  4. (4)

assuming that ${{\boldsymbol{k}}}$ comes equipped with a complete absolute
value, the system $\bigl{(}{{\boldsymbol{k}}}\\{X_{1},\dots,X_{m}\\}\bigr{)}$,
where ${{\boldsymbol{k}}}\\{X_{1},\dots,X_{m}\\}$ consists of all power series
in ${{\boldsymbol{k}}}[[X_{1},\dots,X_{m}]]$ that converge in some
neighborhood of the origin. (See, e.g., [26, Kapitel I, §4].)

In the rest of this section we let $W=(W_{m})$ be a Weierstrass system over
${{\boldsymbol{k}}}$. Note that the definition of Weierstrass system given
above implicitly depends on our fixed choice of the sequence $(X_{m})$ of
indeterminates. If $Y=(Y_{1},\dots,Y_{m})$ is an arbitrary tuple of distinct
indeterminates, then we let ${{\boldsymbol{k}}}\lfloor
Y\rfloor:=\big{\\{}f\in{{\boldsymbol{k}}}[[Y]]:f(X)\in W_{m}\big{\\}}$. Here
as before $X=(X_{1},\dots,X_{m})$. We also denote by $(X)$ the ideal of
$W_{m}={{\boldsymbol{k}}}\lfloor X\rfloor$ generated by $X_{1},\dots,X_{m}$.
In the rest of this subsection we also let $Y=(Y_{1},\dots,Y_{n})$ be a tuple
of distinct indeterminates disjoint from $X$.

With these conventions, we now list some basic facts about Weierstrass systems
which we will be using. First, by [17, Remarks 1.3(2)] we have:

######  Lemma 2.2.

Let $f\in{{\boldsymbol{k}}}\lfloor X,Y\rfloor$, $g\in(X)^{n}$. Then
$f(X,g(X))\in{{\boldsymbol{k}}}\lfloor X\rfloor$.

###### Proof.

For later use we include the proof. With $g=(g_{1},\dots,g_{n})$, repeated
application of (W5) in ${{\boldsymbol{k}}}\lfloor X,Y\rfloor$ yields

$\displaystyle f(X,Y)$
$\displaystyle=u_{n}(X,Y)(Y_{n}-g_{n}(X))+R_{1}(X,Y_{1},\dots,Y_{n-1})$
$\displaystyle=u_{n}(X,Y)(Y_{n}-g_{n}(X))+u_{n-1}(X,Y)(Y_{n-1}-g_{n-1}(X))$
$\displaystyle\qquad\qquad+R_{2}(X,Y_{1},\dots,Y_{n-2})$
$\displaystyle\qquad\vdots$
$\displaystyle=u_{n}(X,Y)(Y_{n}-g_{n}(X))+\cdots+u_{1}(X,Y)(Y_{1}-g_{1}(X))+R_{n}(X)$

and hence $f(X,g(X))=R_{n}(X)\in{{\boldsymbol{k}}}\lfloor X\rfloor$. ∎

######  Corollary 2.3.

For all $f\in{{\boldsymbol{k}}}\lfloor Y\rfloor$ and $g\in(X)^{n}$ we have
$f(g(X))\in{{\boldsymbol{k}}}\lfloor X\rfloor$.

###### Proof.

By (W3) we have

$f^{\prime}:=f(X_{1},\dots,X_{n})\in{{\boldsymbol{k}}}\lfloor
X_{1},\dots,X_{m+n}\rfloor.$

Let $\sigma$ be a permutation of $\\{1,\dots,m+n\\}$ with $\sigma(i)=m+i$ for
$i=1,\dots,n$; then

$h:=f^{\prime}(X_{\sigma(1)},\dots,X_{\sigma(m+n)})\in{{\boldsymbol{k}}}\lfloor
X_{1},\dots,X_{m+n}\rfloor$

by (W2). So by Lemma 2.2 applied to $h$ in place of $f$ and
$Y=(X_{m+1},\dots,X_{m+n})$ we get
$f(g(X))=h(X,g(X))\in{{\boldsymbol{k}}}\lfloor X\rfloor$. ∎

By [17, Remarks 1.3(1),(7),(8)], we have:

######  Lemma 2.4.

The ring ${{\boldsymbol{k}}}\lfloor X\rfloor$ is noetherian regular local; its
unique maximal ideal is $(X)$, and its completion is
${{\boldsymbol{k}}}[[X]]$.

Together with [42, Theorem 8.14] this yields:

######  Corollary 2.5.

The ring ${{\boldsymbol{k}}}[[X]]$ is a faithfully flat
${{\boldsymbol{k}}}\lfloor X\rfloor$-module.

The following is [17, Remarks 1.3(4)]:

######  Lemma 2.6 (Implicit Function Theorem).

Let $f=(f_{1},\dots,f_{n})\in{{\boldsymbol{k}}}\lfloor X,Y\rfloor^{n}$ be such
that

$f\equiv 0\mod(X,Y)\quad\text{and}\quad\det\left(\partial f_{i}/\partial
Y_{j}\right)\not\equiv 0\mod(X,Y).$

Then there is some $y=(y_{1},\dots,y_{n})\in(X)^{n}$ such that $f(X,y)=0$.

Using a special case of the previous lemma we obtain:

######  Lemma 2.7.

The local ring $R={{\boldsymbol{k}}}\lfloor X\rfloor$ is henselian.

###### Proof.

Let $Y$ be a single indeterminate over $R$ and

$P(Y)=1+Y+ab_{2}Y^{2}+\cdots+ab_{d}Y^{d}\in R[Y]$

where $d\geqslant 2$, $a\in(X)$, $b_{2},\dots,b_{d}\in R$. It suffices to show
that there is some $y\in R$ such that $P(y)=0$. To see this consider
$f(X,Y):=P(Y-1)\in R[Y]\subseteq{{\boldsymbol{k}}}\lfloor X,Y\rfloor$. Then
$f\equiv 0\bmod(X,Y)$ and $\partial f/\partial Y\equiv 1\bmod(X,Y)$. Now use
Lemma 2.6. ∎

The local ring ${{\boldsymbol{k}}}[[X]]^{\operatorname{a}}$ is the
henselization of its local subring ${{\boldsymbol{k}}}[X]_{\mathfrak{m}}$
where $\mathfrak{m}$ is the ideal of ${{\boldsymbol{k}}}[X]$ generated by
$X_{1},\dots,X_{m}$. (See, e.g., [35, Théorème 4].) From this fact in
combination with the universal property of henselizations and Lemma 2.7, we
obtain
${{\boldsymbol{k}}}[[X]]^{\operatorname{a}}\subseteq{{\boldsymbol{k}}}\lfloor
X\rfloor$. This is not used later, in the contrast to the following, more
immediate consequence of Lemma 2.7:

######  Corollary 2.8.

Let $f\in{{\boldsymbol{k}}}\lfloor X\rfloor$, $k\geqslant 1$, and suppose
$\operatorname{char}{{\boldsymbol{k}}}=0$ or
$\operatorname{char}{{\boldsymbol{k}}}=p>0$ and $p\\!\not|k$. Then

$\text{$f=g^{k}$ for some $g\in{{\boldsymbol{k}}}\lfloor
X\rfloor^{\times}$}\quad\Longleftrightarrow\quad\text{$f(0)=a^{k}$ for some
$a\in{{\boldsymbol{k}}}^{\times}$.}$

In particular, the group of $1$-units of ${{\boldsymbol{k}}}\lfloor X\rfloor$
is $k$-divisible.

Recall that ${{\boldsymbol{k}}}$ is called euclidean if $a^{2}+b^{2}\neq-1$
for all $a,b\in{{\boldsymbol{k}}}$, and for all $a\in{{\boldsymbol{k}}}$ there
is some $b\in{{\boldsymbol{k}}}$ with $a=b^{2}$ or $a=-b^{2}$. In this case
${{\boldsymbol{k}}}$ has a unique ordering making it an ordered field, given
by $a\geqslant 0\Leftrightarrow a=b^{2}$ for some $b\in{{\boldsymbol{k}}}$,
and below we then always view ${{\boldsymbol{k}}}$ as an ordered field this
way. (For example, each real closed field is euclidean, as is the field of
real constructible numbers; but neither ${\mathbb{Q}}$ nor any algebraic
number field is euclidean [37, Corollary 5.12].)

######  Lemma 2.9.

Suppose ${{\boldsymbol{k}}}$ is euclidean. Let $K$ be an ordered field and
$\sigma\colon{{\boldsymbol{k}}}\lfloor X\rfloor\to K$ be a ring morphism, and
let $u\in{{\boldsymbol{k}}}\lfloor X\rfloor$. Then

$u(0)>0\quad\Longleftrightarrow\quad u\in{{\boldsymbol{k}}}\lfloor
X\rfloor^{\times}\text{ and }\sigma(u)>0.$

###### Proof.

Suppose $u(0)>0$. Then $u\in{{\boldsymbol{k}}}\lfloor X\rfloor^{\times}$, and
since $u(0)$ is a square in ${{\boldsymbol{k}}}$, $u$ is a square in
${{\boldsymbol{k}}}\lfloor X\rfloor$ by Corollary 2.8, so $\sigma(u)>0$.
Conversely, suppose $u\in{{\boldsymbol{k}}}\lfloor X\rfloor^{\times}$ and
$\sigma(u)>0$. Then $u(0)\neq 0$; if $u(0)<0$ then $\sigma(u)<0$, by what we
just showed applied to $-u$ in place of $u$, a contradiction. Thus $u(0)>0$. ∎

######  Lemma 2.10 (Weierstrass Preparation).

Suppose $m\geqslant 1$. Let $g\in{{\boldsymbol{k}}}\lfloor X\rfloor$ be
regular in $X_{m}$ of order $d$, and set $X^{\prime}:=(X_{1},\dots,X_{m-1})$.
Then there are a unit $u$ of ${{\boldsymbol{k}}}\lfloor X\rfloor$ and a
polynomial

$w=X_{m}^{d}+w_{1}X_{m}^{d-1}+\cdots+w_{d}\qquad\text{where
$w_{1},\dots,w_{d}\in(X^{\prime})$}$

such that $g=uw$.

This is [17, Remarks 1.3(9)]. Finally, we note:

######  Lemma 2.11.

Suppose ${{\boldsymbol{k}}}$ is perfect. If $P$ is a prime ideal of
${{\boldsymbol{k}}}\lfloor X\rfloor$, then the ideal
$P{{\boldsymbol{k}}}[[X]]$ of ${{\boldsymbol{k}}}[[X]]$ generated by $P$ is
also prime.

###### Proof.

By [17, Theorem 2.1] in combination with Lemmas 2.4 and 2.7,
${{\boldsymbol{k}}}\lfloor X\rfloor$ is a Weierstrass ring in the sense of
Nagata [45, §45], so we can use [45, (45.1)]. ∎

###  2.2. Infinitesimal $W$-structures

In this subsection we let $K$ be a valued field with valuation ring
$\mathcal{O}=\mathcal{O}_{K}$ and maximal ideal $\smallo=\smallo_{K}$. The
following definition is modeled on [22, axioms C1)–C3) in (2.1)]:

######  Definition 2.12.

An infinitesimal $W$-structure on $K$ is a family $(\phi_{m})_{m\geqslant 0}$
of ring morphisms

$\phi_{m}\colon W_{m}\to\big{\\{}\text{ring of functions
$\smallo^{m}\to\mathcal{O}$}\big{\\}}$

such that:

  * (A1)

$\phi_{m}(X_{i})=\text{$i$-th coordinate function on $\smallo^{m}$}$, for each
$i=1,\dots,m$;

  * (A2)

the map $\phi_{m+1}$ extends $\phi_{m}$, if we identify in the obvious way
functions on $\smallo^{m}$ with functions on $\smallo^{m+1}$ that do not
depend on the last coordinate;

  * (A3)

for all $f\in W_{m}$, permutations $\sigma$ of $\\{1,\dots,m\\}$, and
$a=(a_{1},\dots,a_{m})\in\smallo^{m}$:

$\phi_{m}\bigl{(}f(X_{\sigma(1)},\dots,X_{\sigma(m)})\bigr{)}(a)=\phi_{m}(f)\bigl{(}a_{\sigma(1)},\dots,a_{\sigma(m)}\bigr{)}.$

Note that $\smallo^{0}=\\{\text{pt}\\}$ is a singleton, and the map which
sends $\alpha\colon\smallo^{0}\to\mathcal{O}$ to the element
$\alpha(\text{pt})$ of $\mathcal{O}$ is an isomorphism from the ring of
functions $\smallo^{0}\to\mathcal{O}$ to the ring $\mathcal{O}$. We identify
this ring with $\mathcal{O}$ via the isomorphism
$\alpha\mapsto\alpha(\text{pt})$. Then $\phi_{0}$ is a ring embedding
${{\boldsymbol{k}}}\to\mathcal{O}$. Hence the valued field $K$ is necessarily
of equicharacteristic $\operatorname{char}{{\boldsymbol{k}}}$. Moreover, if
$f\in{{\boldsymbol{k}}}[X]\subseteq W_{m}$, then
$\phi_{m}(f)\colon\smallo^{m}\to\mathcal{O}$ is given by $a\mapsto f(a)$.

######  Lemma 2.13.

For an infinitesimal $W$-structure $(\phi_{m})$ on $K$ and
$f\in{{\boldsymbol{k}}}\lfloor X\rfloor$:

$\phi_{m}(f)=\phi_{0}(f(0))+\phi_{m}(g)\quad\text{ where $g\in(X)$;}$

in particular $\phi_{m}(f)(\smallo^{m})\subseteq\phi_{0}(f(0))+\smallo$ and
$\phi_{m}(f)(0)=\phi_{0}(f(0))$.

This follows immediately from (A1), (A2), and the fact that $f-f(0)\in(X)$.

######  Corollary 2.14.

Suppose $\mathcal{O}=K$, and let $\iota\colon{{\boldsymbol{k}}}\to K$ be an
embedding. Then there is a unique infinitesimal $W$-structure $(\phi_{m})$ on
$K$ such that $\iota=\phi_{0}$.

###### Proof.

We have $\smallo=\\{0\\}$, therefore the family $(\phi_{m})$, where
$\phi_{m}(f)$ is the function $\\{0\\}^{m}\to K$ with value $\iota(f(0))\in
K$, for $f\in W_{m}$, is an infinitesimal $W$-structure on $K$ with
$\iota=\phi_{0}$. By the previous lemma this is also the only infinitesimal
$W$-structure $(\phi_{m})$ on $K$ with $\iota=\phi_{0}$. ∎

Let $K_{0}$ be a valued subfield of $K$, with valuation ring $\mathcal{O}_{0}$
and maximal ideal $\smallo_{0}$; so $\mathcal{O}_{0}=\mathcal{O}\cap K_{0}$
and $\smallo_{0}=\smallo\cap K_{0}$. We say that the infinitesimal
$W$-structure $(\phi_{m})$ on $K$ restricts to an infinitesimal $W$-structure
on $K_{0}$ if $\phi_{m}(\smallo_{0}^{m})\subseteq\mathcal{O}_{0}$ for each
$m$. In this case, the family $(\psi_{m})$, where for each $m$ and $f\in
W_{m}$ we let $\psi_{m}(f)$ be the restriction of
$\phi_{m}(f)\colon\smallo^{m}\to\mathcal{O}$ to a map
$\smallo_{0}^{m}\to\mathcal{O}_{0}$, is an infinitesimal $W$-structure on
$K_{0}$. We call $(\psi_{m})$ the restriction of the $W$-structure on $K$ to
$K_{0}$.

######  Examples.

Here are a few examples of infinitesimal $W$-structures:

  1. (1)

Let $\Gamma$ be an ordered abelian group, written additively, and let
$t^{\Gamma}$ be a multiplicative copy of $\Gamma$, with isomorphism
$\gamma\mapsto t^{\gamma}\colon\Gamma\to t^{\Gamma}$. Let
${{\boldsymbol{k}}}(\\!(t^{\Gamma})\\!)$ be the field of Hahn series with
coefficients in ${{\boldsymbol{k}}}$ and monomials in $t^{\Gamma}$. Its
elements are the formal series

$f=\sum_{\gamma\in\Gamma}f_{\gamma}t^{\gamma}\qquad(f_{\gamma}\in{{\boldsymbol{k}}})$

whose support $\operatorname{supp}f:=\\{\gamma\in\Gamma:f_{\gamma}\neq 0\\}$
is a well-ordered subset of $\Gamma$, added and multiplied in the natural way.
The field ${{\boldsymbol{k}}}(\\!(t^{\Gamma})\\!)$ carries a valuation
$v\colon{{\boldsymbol{k}}}(\\!(t^{\Gamma})\\!)^{\times}\to\Gamma$ given by

$f\mapsto v(f):=\min\operatorname{supp}f,$

which we call the $t$-adic valuation on
${{\boldsymbol{k}}}(\\!(t^{\Gamma})\\!)$. Its valuation ring is

$\mathcal{O}=\big{\\{}f\in{{\boldsymbol{k}}}(\\!(t^{\Gamma})\\!):\operatorname{supp}f\geqslant
0\big{\\}}.$

The kernel of the surjective ring morphism $f\mapsto
f_{0}\colon\mathcal{O}\to{{\boldsymbol{k}}}$ is the maximal ideal

$\smallo=\big{\\{}f\in{{\boldsymbol{k}}}(\\!(t^{\Gamma})\\!):\operatorname{supp}f>0\big{\\}}$

of $\mathcal{O}$, so this morphism induces an isomorphism from the residue
field $\mathcal{O}/\smallo$ of $\mathcal{O}$ onto the coefficient field
${{\boldsymbol{k}}}$. For $f\in{{\boldsymbol{k}}}[[X]]$ and $a\in\smallo^{m}$,
the series $f(a)$ makes sense in ${{\boldsymbol{k}}}(\\!(t^{\Gamma})\\!)$. For
$f\in W_{m}$ let $\phi_{m}(f)$ be the function $a\mapsto
f(a)\colon\smallo^{m}\to\mathcal{O}$. Then the family $(\phi_{m})$ is an
infinitesimal $W$-structure on ${{\boldsymbol{k}}}(\\!(t^{\Gamma})\\!)$ where
$\phi_{0}\colon{{\boldsymbol{k}}}\to\mathcal{O}$ is the natural embedding.

  2. (2)

Equip the field

${{\boldsymbol{k}}}(\\!(t^{*})\\!):=\bigcup_{d\geqslant
1}{{\boldsymbol{k}}}(\\!(t^{1/d})\\!)$

of Puiseux series over ${{\boldsymbol{k}}}$ with the $t$-adic valuation ring

${{\boldsymbol{k}}}[[t^{*}]]:=\bigcup_{d\geqslant
1}{{\boldsymbol{k}}}[[t^{1/d}]],$

with maximal ideal

$\bigcup_{d\geqslant 1}t^{1/d}{{\boldsymbol{k}}}[[t^{1/d}]].$

This is a valued subfield of the Hahn field
${{\boldsymbol{k}}}(\\!(t^{{\mathbb{Q}}})\\!)$. The infinitesimal
$W$-structure on ${{\boldsymbol{k}}}(\\!(t^{{\mathbb{Q}}})\\!)$ described in
(1) restricts to one on ${{\boldsymbol{k}}}(\\!(t^{*})\\!)$.

  3. (3)

Suppose $\Gamma$ is archimedean. Then the completion of the
${{\boldsymbol{k}}}$-subalgebra ${{\boldsymbol{k}}}[t^{\Gamma}]$ of
${{\boldsymbol{k}}}(\\!(t^{\Gamma})\\!)$ under the $t$-adic valuation is the
subfield

$\hskip
50.00008pt\operatorname{cl}({{\boldsymbol{k}}}[t^{\Gamma}]):=\left\\{f\in{{\boldsymbol{k}}}(\\!(t^{\Gamma})\\!)\,:\,\parbox{180.00027pt}{$\operatorname{supp}f$
is finite or $\operatorname{supp}f=\\{\gamma_{0},\gamma_{1},\dots\\}$ with
$\gamma_{n}\to\infty$ as $n\to\infty$}\right\\}$

of ${{\boldsymbol{k}}}(\\!(t^{\Gamma})\\!)$. (See [3, Example 3.2.19].) We
have
${{\boldsymbol{k}}}(\\!(t^{*})\\!)\subseteq\operatorname{cl}({{\boldsymbol{k}}}[t^{\mathbb{Q}}])$.
The infinitesimal $W$-structure on
${{\boldsymbol{k}}}(\\!(t^{{\mathbb{Q}}})\\!)$ from (1) restricts to one on
$\operatorname{cl}({{\boldsymbol{k}}}[t^{\Gamma}])$.

  4. (4)

Suppose ${{\boldsymbol{k}}}$ comes equipped with a complete absolute value,
and let $K:={{\boldsymbol{k}}}\\{\\!\\{t^{*}\\}\\!\\}$ be the subfield of
${{\boldsymbol{k}}}(\\!(t^{*})\\!)$ consisting of the convergent Puiseux
series over ${{\boldsymbol{k}}}$, equipped with the valuation ring
$\mathcal{O}_{t}:={{\boldsymbol{k}}}[[t^{*}]]\cap K$, with maximal ideal
$\smallo_{t}$. Suppose
$W=\bigl{(}{{\boldsymbol{k}}}\\{X_{1},\dots,X_{m}\\}\bigr{)}$ is the
Weierstrass system of convergent power series rings. Then for
$f\in{{\boldsymbol{k}}}\\{X_{1},\dots,X_{m}\\}$ and $a\in\smallo_{t}^{m}$ we
have $f(a)\in K$, so the infinitesimal $W$-structure on
${{\boldsymbol{k}}}(\\!(t^{*})\\!)$ from (2) above restricts to an
infinitesimal $W$-structure on $K$.

In the rest of this subsection we let $(\phi_{m})$ be an infinitesimal
$W$-structure on $K$. We always identify ${{\boldsymbol{k}}}$ with a subfield
of $\mathcal{O}$ via the embedding
$\phi_{0}\colon{{\boldsymbol{k}}}\to\mathcal{O}$. Note that then each
$\phi_{m}$ is a ${{\boldsymbol{k}}}$-algebra morphism from
$W_{m}={{\boldsymbol{k}}}\lfloor X\rfloor$ to the ${{\boldsymbol{k}}}$-algebra
of functions $\smallo^{m}\to\mathcal{O}$. Thus by Lemma 2.13, for $f\in W_{m}$
we have $\phi_{m}(f)(0)=f(0)$. If $Y=(Y_{1},\dots,Y_{m})$ is an arbitrary
tuple of distinct indeterminates, then for $f=F(Y)\in{{\boldsymbol{k}}}\lfloor
Y\rfloor$ we let $\phi_{m}(f):=\phi_{m}(f(X))$. In the next lemma and its
corollary $Y=(Y_{1},\dots,Y_{n})$ is a tuple of distinct indeterminates
disjoint from $X=(X_{1},\dots,X_{m})$.

######  Lemma 2.15 (Substitution).

Let $f\in{{\boldsymbol{k}}}\lfloor X,Y\rfloor$ and
$g=(g_{1},\dots,g_{n})\in{{\boldsymbol{k}}}\lfloor X\rfloor^{n}$ with
$g(0)=0$. Then $f(X,g(X))\in{{\boldsymbol{k}}}\lfloor X\rfloor$ and

$\phi_{m}\bigl{(}f(X,g(X))\bigr{)}(a)=\phi_{m+n}(f)\bigl{(}a,\phi_{m}(g)(a)\bigr{)}\qquad\text{for
all $a\in\smallo^{m}$.}$

###### Proof.

Lemma 2.2 shows $f(X,g(X))\in{{\boldsymbol{k}}}\lfloor X\rfloor$. As in the
proof of that lemma take $u_{1},\dots,u_{n}\in{{\boldsymbol{k}}}\lfloor
X,Y\rfloor$ and $R\in{{\boldsymbol{k}}}\lfloor X\rfloor$ such that

$f(X,Y)=u_{n}(X,Y)(Y_{n}-g_{n}(X))+\cdots+u_{1}(X,Y)(Y_{1}-g_{1}(X))+R(X).$

Then $f(X,g(X))=R(X)$ and so

$\phi_{m}\big{(}f(X,g(X))\big{)}=\phi_{m}(R)=\phi_{m+n}(R)$

by (A2). Let $a\in\smallo^{m}$ and set $b:=\phi_{m}(g)(a)$; by (A2) we also
have $b=\phi_{m+n}(g)(a,b)$ and so using (A1),

$\phi_{m+n}(f)\bigl{(}a,\phi_{m}(g)(a)\bigr{)}=\phi_{m+n}(R)(a,b)=\phi_{m}\bigl{(}f(X,g(X))\bigr{)}(a).\qed$

######  Corollary 2.16.

Let $f\in{{\boldsymbol{k}}}\lfloor Y\rfloor$ and
$g=(g_{1},\dots,g_{n})\in{{\boldsymbol{k}}}\lfloor X\rfloor^{n}$ with
$g(0)=0$. Then

$\phi_{m}\bigl{(}f(g(X))\bigr{)}=\phi_{n}(f)\circ\phi_{m}(g).$

###### Proof.

Let $f^{\prime}$, $\sigma$, $h$ be as in the proof of Corollary 2.3. Then for
$a\in\smallo^{m}$ we have

$\displaystyle\phi_{m}\bigl{(}f(g(X))\bigr{)}(a)$
$\displaystyle=\phi_{m}\bigl{(}h(X,g(X))\bigr{)}(a)$
$\displaystyle=\phi_{m+n}(h)\big{(}a,\phi_{m}(g)(a)\big{)}$
$\displaystyle=\phi_{m+n}(f^{\prime})\big{(}\phi_{m}(g)(a),b\big{)}\quad\text{for
some $b\in\smallo^{m}$}$
$\displaystyle=\phi_{n}(f)\big{(}\phi_{m}(g)(a)\big{)},$

where we used Lemma 2.15 for the second, (A3) for the third, and (A2) for the
fourth equality. ∎

Let now $d\in{\mathbb{N}}$, $d\geqslant 1$. By Corollary 2.3, the
${{\boldsymbol{k}}}$-algebra automorphism $\tau_{d}$ of
${{\boldsymbol{k}}}[[X]]$ restricts to a ${{\boldsymbol{k}}}$-algebra
automorphism of ${{\boldsymbol{k}}}\lfloor X\rfloor$. Now also consider the
automorphism of the ${{\boldsymbol{k}}}$-linear space $\smallo^{m}$, also
denoted by $\tau_{d}$, given by

$(a_{1},\dots,a_{m})\mapsto\big{(}a_{1}+a_{m}^{d^{m-1}},\dots,a_{m-1}+a_{m}^{d},a_{m}\big{)}.$

By (A1) and the previous corollary, we obtain:

######  Corollary 2.17.

For each $f\in{{\boldsymbol{k}}}\lfloor X\rfloor$ and $d\in{\mathbb{N}}$,
$d\geqslant 1$: $\phi_{m}(\tau_{d}(f))=\phi_{m}(f)\circ\tau_{d}$.

To illustrate Corollary 2.17 we prove that if $\mathcal{O}\neq K$, then each
ring morphism $\phi_{m}$ is an embedding; more precisely:

######  Lemma 2.18.

Suppose the valuation of $K$ is non-trivial. Then for each non-zero
$f\in{{\boldsymbol{k}}}\lfloor X\rfloor$ and non-empty open
$U\subseteq\smallo^{m}$ there is some $a\in U$ with $\phi_{m}(f)(a)\neq 0$.

###### Proof.

The case $m=0$ was already observed, so suppose $m\geqslant 1$. Take
$\tau=\tau_{d}$ ($d\in{\mathbb{N}}$, $d\geqslant 1$) such that $\tau(f)$ is
regular in $X_{m}$. Replacing $f$, $U$ by $\tau(f)$, $\tau^{-1}(U)$ and using
Corollary 2.17 we arrange that $f$ is regular in $X_{m}$. Lemma 2.10 yields
$u\in{{\boldsymbol{k}}}\lfloor X\rfloor^{\times}$ and a monic
$w\in{{\boldsymbol{k}}}\lfloor X^{\prime}\rfloor[X_{m}]$, where
$X^{\prime}=(X_{1},\dots,X_{m-1})$, such that $f=uw$. Take
$a^{\prime}\in\smallo^{m-1}$ with
$U\cap(\\{a^{\prime}\\}\times\smallo)\neq\emptyset$. Since $\smallo$ is
infinite, we get an $a_{m}\in\smallo$ such that $a:=(a^{\prime},a_{m})\in U$
and $\phi_{m}(w)(a)\neq 0$, and then also $\phi_{m}(f)(a)\neq 0$. ∎

From now on we often drop $\phi_{m}$ from the notation: for $a\in\smallo^{m}$,
$f\in{{\boldsymbol{k}}}\lfloor X\rfloor$ we write $f(a)$ instead of
$\phi_{m}(f)(a)$. By (A1) we have $g(\smallo^{m})\subseteq\smallo$ for
$g\in(X)$ and hence $f(\smallo^{m})\subseteq f(0)+\smallo\subseteq\mathcal{O}$
for $f\in{{\boldsymbol{k}}}\lfloor X\rfloor$, by Lemma 2.13. Hence given
$a\in\smallo^{m}$, the map $f\mapsto f(a)\colon{{\boldsymbol{k}}}\lfloor
X\rfloor\to\mathcal{O}$ is a local ring morphism. The presence of an
infinitesimal $W$-structure on a valued field imposes a significant
constraint:

######  Lemma 2.19.

The valued field $K$ is henselian.

###### Proof.

Let $a_{1},\dots,a_{m}\in\smallo$ and
$P(Y):=1+Y+a_{1}Y^{2}+\cdots+a_{m}Y^{m+1}$ where $Y$ is a single new
indeterminate. It suffices to show that $P$ has a zero in $\mathcal{O}$. Put

$Q(X,Y):=1+Y+X_{1}Y^{2}+\cdots+X_{m}Y^{m+1}\in{{\boldsymbol{k}}}\lfloor
X\rfloor[Y].$

Since ${{\boldsymbol{k}}}\lfloor X\rfloor$ is henselian (Lemma 2.7), there is
some $f\in{{\boldsymbol{k}}}\lfloor X\rfloor$ with $Q(X,f(X))=0$. Then
$P(b)=Q(a,b)=0$ for $a:=(a_{1},\dots,a_{m})$ and $b:=f(a)\in\mathcal{O}$. ∎

This way we reprove the well-known fact that the valued field
${{\boldsymbol{k}}}(\\!(t^{\Gamma})\\!)$ and its valued subfield
$\operatorname{cl}({{\boldsymbol{k}}}[t^{\Gamma}])$ (when $\Gamma$ is
archimedean) are henselian. Similarly, so are the valued fields
${{\boldsymbol{k}}}(\\!(t^{*})\\!)$ and
${{\boldsymbol{k}}}\\{\\!\\{t^{*}\\}\\!\\}$ (when ${{\boldsymbol{k}}}$ is
equipped with a complete absolute value).

######  Lemma 2.20.

Let $L$ be a valued field extension of $K$ equipped with an infinitesimal
$W$-structure which restricts to that of $K$, and let $K_{0}$ be a valued
subfield of $L$ containing $K$ which is algebraic over $K$. Then the
infinitesimal $W$-structure of $L$ restricts to an infinitesimal $W$-structure
on $K_{0}$.

###### Proof.

By induction on $n$ we show: if $f\in{{\boldsymbol{k}}}\lfloor X,Y\rfloor$
where $Y=(Y_{1},\dots,Y_{n})$ is a tuple of distinct indeterminates disjoint
from $X$ and $b\in\smallo^{m}$, $c\in\smallo_{K_{0}}^{n}$, then $f(b,c)\preceq
1$. For $n=0$ this is clear, so suppose $n\geqslant 1$. Let

$P=Z^{d}+a_{1}Z^{d-1}+\cdots+a_{d}\in K_{0}[Z]\qquad(a_{1},\dots,a_{d}\in K)$

be the minimum polynomial of $c_{n}$ over $K$. Then
$a:=(a_{1},\dots,a_{d})\in\smallo^{d}$ since $K$ is henselian and $c_{n}\prec
1$. We have $g(a,c_{n})=0$ where

$g(U,Y_{n}):=Y_{n}^{d}+U_{1}Y_{n}^{d-1}+\cdots+U_{d}\in{\mathbb{Z}}[U,Y_{n}]\subseteq{{\boldsymbol{k}}}\lfloor
U,Y\rfloor\quad(U=(U_{1},\dots,U_{d})).$

Let $Y^{\prime}=(Y_{1},\dots,Y_{n-1})$. Weierstrass Division in
${{\boldsymbol{k}}}\lfloor U,X,Y\rfloor$ yields $q\in{{\boldsymbol{k}}}\lfloor
U,X,Y\rfloor$ and $r\in{{\boldsymbol{k}}}\lfloor U,X,Y^{\prime}\rfloor[Y_{n}]$
of degree $<d$ with $f=qg+r$. Then $f(b,c)=r(a,b,c)\preceq 1$ by the inductive
hypothesis applied to the coefficients of $r$ and $(a,b)$, $c^{\prime}$ in
place of $b$, $c$, respectively. ∎

Let ${{\boldsymbol{k}}}^{\operatorname{a}}$ be an algebraic closure of
${{\boldsymbol{k}}}$; then
${{\boldsymbol{k}}}^{\operatorname{a}}(\\!(t^{\mathbb{Q}})\\!)$ is
algebraically closed. By the previous lemma, the infinitesimal $W$-structure
of ${{\boldsymbol{k}}}^{\operatorname{a}}(\\!(t^{\mathbb{Q}})\\!)$ restricts
to an infinitesimal $W$-structure on the algebraic closure of the field
${{\boldsymbol{k}}}(\\!(t)\\!)$ of Laurent series over ${{\boldsymbol{k}}}$
inside ${{\boldsymbol{k}}}^{\operatorname{a}}(\\!(t^{\mathbb{Q}})\\!)$.

###  2.3. Infinitesimal $W$-structures as model-theoretic structures

Let $\mathcal{L}$ be an expansion of the language
$\mathcal{L}_{\preceq}=\\{0,1,{-},{+},{\,\cdot\,},{\preceq}\\}$ of rings
augmented by a binary relation symbol $\preceq$. (See Section 1.) We then let
$\mathcal{L}_{W}$ be the expansion of $\mathcal{L}$ by a new $m$-ary function
symbol, for each $m$ and $f\in W_{m}$, also denoted by $f$. Given an
$\mathcal{L}$-theory $T$ containing the $\mathcal{L}_{\preceq}$-theory of
valued fields we let $T_{W}$ be the $\mathcal{L}_{W}$-theory whose models are
the $\mathcal{L}_{W}$-structures $\boldsymbol{K}=(K;\dots)$ whose
$\mathcal{L}$-reduct is a model of $T$, each $f\in W_{m}$ is interpreted by a
function ${f^{\boldsymbol{K}}\colon K^{m}\to K}$ which is identically zero on
the complement of $\smallo^{m}$ and satisfies
$f^{\boldsymbol{K}}(\smallo^{m})\subseteq\mathcal{O}$, and such that the
family $(\phi_{m})$ of maps given by
$\phi_{m}(f)=f^{\boldsymbol{K}}|{\smallo^{m}}$ is an infinitesimal
$W$-structure on $K$. (Note: the underlying valued field of each model of
$T_{W}$ is henselian of equicharacteristic
$\operatorname{char}{{\boldsymbol{k}}}$.)

######  Example.

Let $q\in{\mathbb{Q}}^{>}$. We then have an automorphism $a(t)\mapsto
a(t^{q})$ of the valued field $K={{\boldsymbol{k}}}(\\!(t^{{\mathbb{Q}}})\\!)$
over ${{\boldsymbol{k}}}$ (with inverse $b(t)\mapsto b(t^{1/q})$). For
$\mathcal{L}=\mathcal{L}_{\preceq}$, this is also an automorphism of the
$\mathcal{L}_{W}$-structure $\boldsymbol{K}=(K;\dots)$ with
$\mathcal{L}$-reduct $K$ described above. This automorphism restricts to an
automorphism of the substructure
$\big{(}{{\boldsymbol{k}}}(\\!(t^{*})\\!);\dots\big{)}$ of $\boldsymbol{K}$.
Given $a\in{{\boldsymbol{k}}}(\\!(t^{*})\\!)$ there is some $d\geqslant 1$
such that $a(t^{d})\in{{\boldsymbol{k}}}(\\!(t)\\!)$, and then

$a\preceq 1\ \Longleftrightarrow\ a(t^{d})\in{{\boldsymbol{k}}}[[t]],\qquad
a\prec 1\ \Longleftrightarrow\ a(t^{d})\in t{{\boldsymbol{k}}}[[t]].$

Suppose ${{\boldsymbol{k}}}$ comes equipped with a complete absolute value and
$W$ is the Weierstrass system of convergent power series over
${{\boldsymbol{k}}}$. Then the automorphism $a(t)\mapsto a(t^{q})$ of the
$\mathcal{L}_{W}$-structure
$\big{(}{{\boldsymbol{k}}}(\\!(t^{*})\\!);\dots\big{)}$ further restricts to
an automorphism of its substructure
$\big{(}{{\boldsymbol{k}}}\\{\\!\\{t^{*}\\}\\!\\};\dots\big{)}$.

###  2.4. Zero sets and vanishing ideals

Let $(\phi_{m})$ be an infinitesimal $W$-structure on a valued field $K$, and
let $I$ be an ideal of $W_{m}={{\boldsymbol{k}}}\lfloor X\rfloor$. We let

$\operatorname{Z}_{K}(I):=\big{\\{}a\in\smallo^{m}:\text{$f(a)=0$ for all
$f\in I$}\big{\\}}$

be the zero set of $I$ in $K$. One verifies easily that if $J$ is another
ideal of ${{\boldsymbol{k}}}\lfloor X\rfloor$, then
$\operatorname{Z}_{K}(I)\subseteq\operatorname{Z}_{K}(J)$ if $I\supseteq J$,
and

$\operatorname{Z}_{K}(IJ)=\operatorname{Z}_{K}(I\cap
J)=\operatorname{Z}_{K}(I)\cup\operatorname{Z}_{K}(J),\qquad\operatorname{Z}_{K}(I+J)=\operatorname{Z}_{K}(I)\cap\operatorname{Z}_{K}(J).$

Let also $S$ be a subset of $\smallo^{m}$. We let

$\operatorname{I}(S):=\big{\\{}f\in{{\boldsymbol{k}}}\lfloor
X\rfloor:\text{$f(a)=0$ for all $a\in S$}\big{\\}}.$

Then $\operatorname{I}(S)$ is an ideal of ${{\boldsymbol{k}}}\lfloor
X\rfloor$, called the vanishing ideal of $S$. If also $T\subseteq\smallo^{n}$,
then $\operatorname{I}(S)\subseteq\operatorname{I}(T)$ if $S\supseteq T$, and
$\operatorname{I}(S\cup T)=\operatorname{I}(S)\cap\operatorname{I}(T)$.
Clearly the ideal $\operatorname{I}(S)$ is radical, so
$\operatorname{I}\\!\big{(}\\!\operatorname{Z}_{K}(I)\big{)}\supseteq\sqrt{I}$.
Also note $\operatorname{Z}_{K}(I)=\operatorname{Z}_{K}(\sqrt{I})$.

###  2.5. Germs of analytic functions

Suppose ${{\boldsymbol{k}}}$ is equipped with a complete absolute value.
Recall that analytic functions $f\colon U\to{{\boldsymbol{k}}}$ and $g\colon
V\to{{\boldsymbol{k}}}$, where $U,V\subseteq{{\boldsymbol{k}}}^{m}$ are open
neighborhoods of $0$, are said to have the same germ at $0$ if there is an
open neighborhood $W$ of $0$ contained in $U\cap V$ such that $f(a)=g(a)$ for
all $a\in W$. The relation of having the same germ at $0$ is an equivalence
relation on the collection of all analytic functions $f\colon
U\to{{\boldsymbol{k}}}$ where $U\subseteq{{\boldsymbol{k}}}^{m}$ is an open
neighborhood of $0$; we denote the equivalence class of $f$ by $[f]$. The
equivalence classes of this equivalence relation form a commutative ring such
that $[f]+[g]=[f+g]$ and $[f]\cdot[g]=[f\cdot g]$ for all analytic functions
$f,g\colon U\to{{\boldsymbol{k}}}$ defined on an open neighborhood
$U\subseteq{{\boldsymbol{k}}}^{m}$ of $0$. The map sending the germ of $f$ to
its Taylor series at $0$ is a ring isomorphism from this ring onto
${{\boldsymbol{k}}}\\{X\\}={{\boldsymbol{k}}}\\{X_{1},\dots,X_{m}\\}$, via
which we identify these two rings.

##  3\. A Proof of Rückert’s Nullstellensatz

In this section $\mathcal{L}=\mathcal{L}_{\preceq}$. We let
$\operatorname{ACVF}$ be the $\mathcal{L}$-theory whose models are the
algebraically closed non-trivially valued fields. We recall a fundamental
theorem of the model theory of algebraically closed valued fields. (See, for
example, [3, Theorem 3.6.1] or [46, Theorem 4.4.2] for a proof.)

######  Theorem 3.1 (A. Robinson [54]).

$\operatorname{ACVF}$ has QE.

In particular, the theory $\operatorname{ACVF}$ is substructure complete, that
is, if $K,L\models\operatorname{ACVF}$ and $A$ is a substructure of both $K$
and $L$, then $K_{A}\equiv L_{A}$. Moreover, $\operatorname{ACVF}$ is the
model completion of the $\mathcal{L}$-theory of pairs $(R,{\preceq})$ where
$R$ is an integral domain and $\preceq$ is a dominance relation on $R$ [3,
Corollary 3.6.4].

In the following we let $W$ be a Weierstrass system over a field
${{\boldsymbol{k}}}$. We also let $X=(X_{1},\dots,X_{m})$ be a tuple of
distinct indeterminates and $y=(y_{1},\dots,y_{n})$ be a tuple of distinct
$\mathcal{L}$-variables. Let $K\models\operatorname{ACVF}_{W}$ and
$\mathcal{O}=\mathcal{O}_{K}$, $\smallo=\smallo_{K}$. Recall that for
$a\in\smallo^{m}$, the map $f\mapsto f(a)$ is a local ring morphism
${{{\boldsymbol{k}}}\lfloor X\rfloor\to\mathcal{O}}$, and that we view
${{\boldsymbol{k}}}$ as a subfield of $\mathcal{O}$. We now use Theorem 3.1 to
prove the following specialization property:

######  Proposition 3.2.

Let $\varphi(y)$ be an $\mathcal{L}$-formula,
$f=(f_{1},\dots,f_{n})\in{{\boldsymbol{k}}}\lfloor X\rfloor^{n}$,
$\Omega\models\operatorname{ACVF}$, and $\sigma\colon{{\boldsymbol{k}}}\lfloor
X\rfloor\to\mathcal{O}_{\Omega}$ be a local ring morphism such that
$\Omega\models\varphi(\sigma(f))$. Then $K\models\varphi(f(a))$ for some
$a\in\smallo^{m}$.

###### Proof.

We proceed by induction on the length $m$ of $X$. Note that if we equip
${{\boldsymbol{k}}}$ with the trivial dominance relation, then the restriction
of $\sigma$ to an embedding ${{\boldsymbol{k}}}\to\Omega$ as well as the
natural inclusion ${{\boldsymbol{k}}}\to K$ are $\mathcal{L}$-embeddings.
Hence the case $m=0$ follows from substructure completeness of
$\operatorname{ACVF}$. Suppose $m\geqslant 1$. By Theorem 3.1 we arrange that
$\varphi$ is quantifier-free, and we may then assume that $\varphi$ has the
form

$\bigwedge_{i\in I}P_{i}(y)=0\wedge Q(y)\neq 0\wedge\bigwedge_{j\in
J}R_{j}(y)\,\square_{j}\,S_{j}(y)$

where $I$, $J$ are finite index sets,
$P_{i},Q,R_{j},S_{j}\in{\mathbb{Z}}[Y_{1},\dots,Y_{n}]$, and each
$\square_{j}$ is one of the symbols $\preceq$ or $\prec$. Now note that for
each $P\in{\mathbb{Z}}[Y_{1},\dots,Y_{n}]$ we have
$P(f)\in{{\boldsymbol{k}}}\lfloor X\rfloor$ as well as
$P(\sigma(f))=\sigma(P(f))$ and $P(f(a))=P(f)(a)$ for every $a\in\smallo^{m}$.
Hence by suitably modifying $\varphi$, $f$ (which may include increasing $n$)
we can arrange that the polynomials $P_{i}$, $Q$, $R_{j}$, $S_{j}$ are just
distinct elements of $\\{Y_{1},\dots,Y_{n}\\}$. For example, for $n=4$, our
formula $\varphi$ may have the form

$y_{1}=0\ \wedge\ y_{2}\neq 0\ \wedge\ y_{3}\,\square\,y_{4}\quad\text{where
$\square$ is either $\preceq$ or $\prec$.}$

(The general case is only notationally more involved.) We may also clearly
arrange that the formal power series $f_{1},\dots,f_{n}$ are all non-zero.
Take $d\in{\mathbb{N}}$, $d\geqslant 1$, such that with $\tau:=\tau_{d}$, each
$\tau(f_{j})$ is regular in $X_{m}$. For each $g\in{{\boldsymbol{k}}}\lfloor
X\rfloor$ and $a\in\smallo^{m}$ we have $\tau(g)(a)=g(\tau(a))$. (Corollary
2.17.) Hence replacing each $f_{j}$ by $\tau(f_{j})$ and $\sigma$ by
$\sigma\circ\tau^{-1}$ we arrange that $f_{j}$ is regular in $X_{m}$, for
$j=1,\dots,n$. Weierstrass Preparation in ${{\boldsymbol{k}}}\lfloor X\rfloor$
(Lemma 2.10) yields a $1$-unit $u_{j}\in{{\boldsymbol{k}}}\lfloor X\rfloor$
and a polynomial $w_{j}\in\mathcal{{{\boldsymbol{k}}}}\lfloor
X^{\prime}\rfloor[X_{m}]$, where $X^{\prime}=(X_{1},\dots,X_{m-1})$, such that
$f_{j}=u_{j}w_{j}$, for $j=1,\dots,n$. We have $\sigma(u_{j})\sim 1$ since
$\sigma$ is local. Hence we may replace each $f_{j}$ by $w_{j}$ to arrange
that $f_{j}=w_{j}$ is a polynomial. Take $e\in{\mathbb{N}}$ and
$w_{jk}\in{{\boldsymbol{k}}}\lfloor X^{\prime}\rfloor$ ($j=1,\dots,n$,
$k=0,\dots,e$) such that

$w_{j}=w_{j0}+w_{j1}X_{m}+\cdots+w_{je}X_{m}^{e},$

and set $w:=(w_{jk})$. Let $u:=(u_{jk})$ be a tuple of distinct new variables
and $v$ be a new variable; then $\sigma(w_{j})=t_{j}(\sigma(w),\sigma(X_{m}))$
for the $\mathcal{L}_{\operatorname{R}}$-term

$t_{j}(u,v):=u_{j0}+u_{j1}v+\cdots+u_{je}v^{e}.$

Consider the existential $\mathcal{L}$-formula

$\psi(u):=\exists v\big{(}v\prec
1\wedge\varphi\big{(}t_{1}(u,v),\dots,t_{n}(u,v)\big{)}\big{)}.$

Then

$\displaystyle\Omega\models\varphi(\sigma(f))$
$\displaystyle\quad\,\Longrightarrow\quad\Omega\models\psi(\sigma(w))$
$\displaystyle\quad\,\Longrightarrow\quad K\models\psi(w(b))\text{ for some
$b\in\smallo^{m-1}$}$ $\displaystyle\quad\Longleftrightarrow\quad
K\models\varphi(f(a))\text{ for some $a\in\smallo^{m}$,}$

where for the second implication we used the inductive hypothesis applied to
$\psi$, $w$, and the restriction of $\sigma$ to ${{\boldsymbol{k}}}\lfloor
X^{\prime}\rfloor$ in place of $\varphi$, $f$, $\sigma$, respectively. ∎

The following corollary (not used later) captures the model-theoretic essence
of Proposition 3.2:

######  Corollary 3.3.

Let $\theta$ be an existential $\mathcal{L}_{W}$-sentence. If
$K\models\theta$, then $L\models\theta$ for each
$L\models\operatorname{ACVF}_{W}$.

###### Proof.

Take a tuple $x=(x_{1},\dots,x_{m})$ of distinct new variables, for some $m$,
and a boolean combination $\psi(x)$ of unnested atomic
$\mathcal{L}_{W}$-formulas with $\models\theta\leftrightarrow\exists x\psi$;
here $\exists x$ abbreviates $\exists x_{1}\cdots\exists x_{m}$. (See [3,
Lemmas B.4.5 and B.5.5].) Then $\psi(x)=\varphi(x,f(x))$ for some quantifier-
free $\mathcal{L}$-formula $\varphi(x,y)$ and $f\in{{\boldsymbol{k}}}\lfloor
X\rfloor^{n}$. Now

$\operatorname{ACVF}_{W}\models\theta\leftrightarrow\exists x\big{(}(1\preceq
x_{1}\vee\cdots\vee 1\preceq x_{m})\wedge\varphi(x,0)\big{)}\vee\exists
x\big{(}x_{1}\prec 1\wedge\cdots\wedge x_{m}\prec 1\wedge\psi\big{)},$

hence the corollary follows from Theorem 3.1 and Proposition 3.2. ∎

######  Corollary 3.4.

Let $\varphi(y)$ be an $\mathcal{L}_{\operatorname{R}}$-formula,
$f\in{{\boldsymbol{k}}}\lfloor X\rfloor^{n}$, and let
$\sigma\colon{{\boldsymbol{k}}}\lfloor X\rfloor\to\Omega$ be a ring morphism
to an algebraically closed field $\Omega$ such that
$\Omega\models\varphi(\sigma(f))$. Then $K\models\varphi(f(a))$ for some
$a\in\smallo^{m}$.

###### Proof.

Suppose first that $\ker\sigma=(X)$; then $\sigma(g)=\sigma(g(0))$ for every
$g\in{{\boldsymbol{k}}}\lfloor X\rfloor$, and the corollary follows from
substructure completeness of the $\mathcal{L}_{\operatorname{R}}$-theory of
algebraically closed fields by taking $a:=0\in\smallo^{m}$. Now suppose
$\ker\sigma\neq(X)$. The image $A$ of $\sigma$ is a local subring of $\Omega$
(cf. Lemma 2.4) but not a field. Take a valuation ring of $\Omega$ lying over
$A$ and equip $\Omega$ with the associated dominance relation. Then
$\Omega\models\operatorname{ACVF}$ and $\sigma\colon{{\boldsymbol{k}}}\lfloor
X\rfloor\to\mathcal{O}_{\Omega}$ is a local ring morphism, so the previous
proposition applies. ∎

Corollary 3.4 now implies a general version of Rückert’s Nullstellensatz:

######  Theorem 3.5 (Rückert’s Nullstellensatz).

Let $I$ be an ideal of ${{\boldsymbol{k}}}\lfloor X\rfloor$. Then

$\operatorname{I}\\!\big{(}\\!\operatorname{Z}_{K}(I)\big{)}=\sqrt{I}.$

###### Proof.

If we have some $a\in\operatorname{Z}_{K}(I)$, then the kernel of the ring
morphism $f\mapsto f(a)\colon{{\boldsymbol{k}}}\lfloor X\rfloor\to K$ is a
prime ideal of ${{\boldsymbol{k}}}\lfloor X\rfloor$ containing $I$. Hence we
may assume that $I\neq{{\boldsymbol{k}}}\lfloor X\rfloor$. Recall: the ring
${{\boldsymbol{k}}}\lfloor X\rfloor$ is noetherian (Lemma 2.4). This yields
prime ideals $P_{1},\dots,P_{k}$ ($k\geqslant 1$) of
${{\boldsymbol{k}}}\lfloor X\rfloor$ such that $\sqrt{I}=P_{1}\cap\cdots\cap
P_{k}$. Then

$\operatorname{I}\\!\big{(}\\!\operatorname{Z}_{K}(I)\big{)}=\operatorname{I}\\!\big{(}\\!\operatorname{Z}_{K}(\sqrt{I})\big{)}=\operatorname{I}\\!\big{(}\\!\operatorname{Z}_{K}(P_{1})\big{)}\cap\cdots\cap\operatorname{I}\\!\big{(}\\!\operatorname{Z}_{K}(P_{k})\big{)},$

so it suffices to treat the case where $I$ is prime. So from now on assume $I$
is a prime ideal of ${{\boldsymbol{k}}}\lfloor X\rfloor$; we need to show that
then $\operatorname{I}\\!\big{(}\\!\operatorname{Z}_{K}(I)\big{)}=I$. Let
$\Omega$ be an algebraic closure of the fraction field of the integral domain
$A:={{\boldsymbol{k}}}\lfloor X\rfloor/I$, and let
$\sigma\colon{{\boldsymbol{k}}}\lfloor X\rfloor\to\Omega$ be the composition
of the residue morphism $f\mapsto f+I\colon{{\boldsymbol{k}}}\lfloor
X\rfloor\to A$ with the natural inclusion $A\subseteq\Omega$. Let
$f\in{{\boldsymbol{k}}}\lfloor X\rfloor\setminus I$ and let
$g_{1},\dots,g_{n}$ generate $I$. Then
$\Omega\models\varphi\big{(}\sigma(f),\sigma(g_{1}),\dots,\sigma(g_{n})\big{)}$
where $\varphi(y_{0},\dots,y_{n})$ is the quantifier-free
$\mathcal{L}_{\operatorname{R}}$-formula $y_{0}\neq 0\wedge
y_{1}=\cdots=y_{n}=0$. Corollary 3.4 yields an $a\in\smallo^{m}$ with
$f(a)\neq 0$ and $g_{1}(a)=\cdots=g_{n}(a)=0$, so
$f\notin\operatorname{I}\\!\big{(}\\!\operatorname{Z}_{K}(I)\big{)}$. Thus
$\operatorname{I}\\!\big{(}\\!\operatorname{Z}_{K}(I)\big{)}\subseteq I$ as
required. ∎

Let now ${{\boldsymbol{k}}}^{\operatorname{a}}$ be an algebraic closure of
${{\boldsymbol{k}}}$ and $K={{\boldsymbol{k}}}(\\!(t)\\!)^{\operatorname{a}}$
be the algebraic closure of ${{\boldsymbol{k}}}(\\!(t)\\!)$ inside
${{\boldsymbol{k}}}^{\operatorname{a}}(\\!(t^{\mathbb{Q}})\\!)$, equipped with
the restriction of the dominance relation of
${{\boldsymbol{k}}}^{\operatorname{a}}(\\!(t^{\mathbb{Q}})\\!)$. Expand $K$ to
an $\mathcal{L}_{W}$-structure where $W$ is the Weierstrass system of formal
power series over ${{\boldsymbol{k}}}$. (See the remarks after Lemma 2.20.)
Then $K\models\operatorname{ACVF}_{W}$, hence by Theorem 3.5 we obtain the
“formal” version of Rückert’s Nullstellensatz:

######  Corollary 3.6.

Let $f,g_{1},\dots,g_{n}\in{{\boldsymbol{k}}}[[X]]$ with ${f(a)=0}$ for all
$a\in\smallo^{m}$ such that $g_{1}(a)=\cdots=g_{n}(a)=0$. Then there are
$h_{1},\dots,h_{n}\in{{\boldsymbol{k}}}[[X]]$ and some $k\geqslant 1$ such
that $f^{k}=g_{1}h_{1}+\cdots+g_{n}h_{n}$.

In the rest of this subsection we assume
$\operatorname{char}{{\boldsymbol{k}}}=0$. Then
$K={{\boldsymbol{k}}}^{\operatorname{a}}(\\!(t^{*})\\!)$ (see, e.g., [3,
Example 3.3.23]). Hence using the automorphisms $a(t)\mapsto a(t^{d})$ of the
$\mathcal{L}_{W}$-structure $K$ (where $d\geqslant 1$), from Corollary 3.6 we
obtain:

######  Corollary 3.7.

Let $f,g_{1},\dots,g_{n}\in{{\boldsymbol{k}}}[[X]]$ with ${f(a)=0}$ for all
$a\in t{{\boldsymbol{k}}}^{\operatorname{a}}[[t]]^{m}$ such that
$g_{1}(a)=\cdots=g_{n}(a)=0$. Then there are power series
$h_{1},\dots,h_{n}\in{{\boldsymbol{k}}}[[X]]$ and some $k\geqslant 1$ such
that $f^{k}=g_{1}h_{1}+\cdots+g_{n}h_{n}$.

For $m=1$ the previous corollary holds without assuming
$\operatorname{char}{{\boldsymbol{k}}}=0$, since ${\smallo\setminus\\{0\\}}$
is an orbit of the group of continuous automorphisms of
${{\boldsymbol{k}}}^{\operatorname{a}}(\\!(t^{\mathbb{Q}})\\!)$ over
${{\boldsymbol{k}}}^{\operatorname{a}}$ [33, Theorem 3.4]. But we do not know
whether this remains true for $m>1$. (See [31, 32] for an explicit description
of the subfield ${{\boldsymbol{k}}}(\\!(t)\\!)^{\operatorname{a}}$ of
${{\boldsymbol{k}}}^{\operatorname{a}}(\\!(t^{\mathbb{Q}})\\!)$.)

######  Remark.

If in the context of Corollary 3.7 we have
$f,g_{1},\dots,g_{n}\in{{\boldsymbol{k}}}[[X]]^{\operatorname{a}}$, then we
can take $h_{1},\dots,h_{n}\in{{\boldsymbol{k}}}[[X]]^{\operatorname{a}}$.
This follows either from Theorem 3.5 applied to the Weierstrass system of
algebraic formal power series, or the previous corollary and Corollary 2.5.
Similarly with ${{\boldsymbol{k}}}[[X]]^{\operatorname{a}}$ replaced by
${{\boldsymbol{k}}}[[X]]^{\operatorname{da}}$.

Now we also suppose that ${{\boldsymbol{k}}}$ is algebraically closed and
comes equipped with a complete absolute value. (For example,
${{\boldsymbol{k}}}={\mathbb{C}}$ or ${{\boldsymbol{k}}}={\mathbb{C}}_{p}$
with their usual absolute values.) Then the field
${{\boldsymbol{k}}}\\{\\!\\{t^{*}\\}\\!\\}$ of convergent Puiseux series over
${{\boldsymbol{k}}}$ is algebraically closed. Arguing as in the proof of
Corollary 3.7, with ${{\boldsymbol{k}}}\\{\\!\\{t^{*}\\}\\!\\}$ in place of
${{\boldsymbol{k}}}(\\!(t^{*})\\!)$ and $W=$ the Weierstrass system of
convergent power series over ${{\boldsymbol{k}}}$, we obtain the classical
version of Rückert’s Nullstellensatz for convergent power series:

######  Corollary 3.8.

Let $f,g_{1},\dots,g_{n}\in{{\boldsymbol{k}}}\\{X\\}$ and suppose that for all
$a\in{{\boldsymbol{k}}}\\{t\\}^{m}$ with ${a(0)=0}$ we have $f(a)=0$ whenever
$g_{1}(a)=\cdots=g_{n}(a)=0$. Then there are
$h_{1},\dots,h_{n}\in{{\boldsymbol{k}}}\\{X\\}$ and some $k\geqslant 1$ such
that $f^{k}=g_{1}h_{1}+\cdots+g_{n}h_{n}$.

As usual this yields a version for germs of analytic functions.

######  Corollary 3.9.

Let $f,g_{1},\dots,g_{n}\colon U\to{{\boldsymbol{k}}}$ be analytic functions,
where $U$ is an open neighborhood of $0$ in ${{\boldsymbol{k}}}^{m}$, such
that for all $a\in U$,

$g_{1}(a)=\cdots=g_{n}(a)=0\quad\Longrightarrow\quad f(a)=0.$

Then there are analytic functions $h_{1},\dots,h_{n}\colon
V\to{{\boldsymbol{k}}}$, for some open neighborhood $V\subseteq U$ of $0$ in
${{\boldsymbol{k}}}^{m}$, and some $k\geqslant 1$ such that

$f^{k}=g_{1}h_{1}+\cdots+g_{n}h_{n}\qquad\text{ on $V$.}$

###### Proof.

Let $f,g_{1},\dots,g_{n}\in{{\boldsymbol{k}}}\\{X\\}$ also denote the germs of
the corresponding analytic functions $U\to{{\boldsymbol{k}}}$. Then for all
$a\in{{\boldsymbol{k}}}\\{t\\}^{m}$ with ${a(0)=0}$ we have
$g_{1}(a)=\cdots=g_{n}(a)=0\Rightarrow f(a)=0$. (By the hypothesis and the
continuity of each representative of $a$ at $0$.) Now Corollary 3.8 yields the
existence of $h_{1},\dots,h_{n}$ and $k$ as claimed. ∎

##  4\. Risler’s Nullstellensatz

Let
$\mathcal{L}=\mathcal{L}_{\operatorname{OR},\preceq}=\\{0,1,{-},{+},{\,\cdot\,},{\leqslant},{\preceq}\\}$
be the language
$\mathcal{L}_{\operatorname{OR}}=\mathcal{L}_{\operatorname{R}}\cup\\{\leqslant\\}$
of ordered rings expanded by a new binary relation symbol $\preceq$. Let
$\operatorname{RCVF}$ be the $\mathcal{L}$-theory whose models are the
$\mathcal{L}$-structures $(K,{\preceq})$ where $K$ is a real closed ordered
field and $\preceq$ is a non-trivial convex dominance relation on $K$. Here, a
dominance relation $\preceq$ on an ordered integral domain $R$ is said to be
convex if $0\leqslant r\leqslant s\Rightarrow r\preceq s$, for all $r,s\in R$.
The analogue of Theorem 3.1 in this context is the following fact. (For a
self-contained proof see [3, Theorem 3.6.6] or [46, Theorem 4.5.1].)

######  Theorem 4.1 (Cherlin-Dickmann [12]).

$\operatorname{RCVF}$ has QE.

Indeed, $\operatorname{RCVF}$ is the model completion of the
$\mathcal{L}$-theory of ordered integral domains equipped with a convex
dominance relation [3, Corollary 3.6.7]. Let $W$ be a Weierstrass system over
a euclidean field ${{\boldsymbol{k}}}$ and $K\models\operatorname{RCVF}_{W}$.
Also let $X=(X_{1},\dots,X_{m})$ be a tuple of distinct indeterminates and
$y=(y_{1},\dots,y_{n})$ be a tuple of distinct $\mathcal{L}$-variables. Just
like Theorem 3.1 gave rise to Proposition 3.2, Theorem 4.1 implies:

######  Proposition 4.2.

Let $\varphi(y)$ be an $\mathcal{L}$-formula,
$f=(f_{1},\dots,f_{n})\in{{\boldsymbol{k}}}\lfloor X\rfloor^{n}$,
$\Omega\models\operatorname{RCVF}$, and $\sigma\colon{{\boldsymbol{k}}}\lfloor
X\rfloor\to\mathcal{O}_{\Omega}$ be a local ring morphism such that
$\Omega\models\varphi(\sigma(f))$. Then $K\models\varphi(f(a))$ for some
$a\in\smallo^{m}$.

###### Proof.

We proceed by induction on $m$ as in the proof of Proposition 3.2. Equipping
${{\boldsymbol{k}}}$ with its unique ordering and with the trivial dominance
relation, the restriction of $\sigma$ to an embedding
${{\boldsymbol{k}}}\to\Omega$ as well as the natural inclusion
${{\boldsymbol{k}}}\to K$ are $\mathcal{L}$-embeddings. Together with
substructure completeness of $\operatorname{RCVF}$ this takes care of the case
$m=0$. So suppose $m\geqslant 1$. Using Theorem 4.1 we arrange $\varphi$ to
have the form

$P(y)=0\wedge\bigwedge_{i\in I}Q_{i}(y)>0\wedge\bigwedge_{j\in
J}R_{j}(y)\,\square_{j}\,S_{j}(y)$

where $I$, $J$ are finite index sets,
$P,Q_{i},R_{j},S_{j}\in{\mathbb{Z}}[Y_{1},\dots,Y_{n}]$, and each
$\square_{j}$ is $\preceq$ or $\prec$. As in the proof of Proposition 3.2 we
also arrange that the $P$, $Q_{i}$, $R_{j}$, $S_{j}$ are distinct elements of
$\\{Y_{1},\dots,Y_{n}\\}$ and $f_{1},\dots,f_{n}$ are all regular in $X_{m}$.
Take some $1$-unit $u_{j}\in{{\boldsymbol{k}}}\lfloor X\rfloor$ and
$w_{j}\in{{\boldsymbol{k}}}\lfloor X^{\prime}\rfloor[X_{m}]$, where
$X^{\prime}=(X_{1},\dots,X_{m-1})$, such that $f_{j}=u_{j}w_{j}$. Then
$\sigma(u_{j})\sim 1$ since $\sigma$ is local, hence $\sigma(u_{j})>0$. Thus
we may arrange that each $f_{j}=w_{j}$ is a polynomial and conclude the
argument as in the proof of Proposition 3.2. ∎

From Proposition 4.2 we obtain the completeness of the existential part of
$\operatorname{RCVR}_{W}$ as in the proof of Corollary 3.3:

######  Corollary 4.3.

For each existential $\mathcal{L}_{W}$-sentence $\theta$, we either have
$\operatorname{RCVR}_{W}\models\theta$ or
$\operatorname{RCVR}_{W}\models\neg\theta$.

Next, an analogue of Corollary 3.4; for this, we first note:

######  Lemma 4.4.

Let $\Omega$ be an ordered field and $\sigma\colon{{\boldsymbol{k}}}\lfloor
X\rfloor\to\Omega$ be a ring morphism. Equip $\Omega$ with the dominance
relation associated to the convex hull of $\sigma({{\boldsymbol{k}}})$ in
$\Omega$. Then for $g\in{{\boldsymbol{k}}}\lfloor X\rfloor$ we have
$\sigma\big{(}g-g(0)\big{)}\prec 1$.

###### Proof.

Let $g\in{{\boldsymbol{k}}}\lfloor X\rfloor$; replacing $g$ by $g-g(0)$ we
arrange $g(0)=0$. Let $\varepsilon\in{{\boldsymbol{k}}}^{>}$ and consider
$u=\varepsilon-g\in{{\boldsymbol{k}}}\lfloor X\rfloor$. Then $\sigma(u)>0$ by
Lemma 2.9 and thus $\sigma(g)<\sigma(\varepsilon)$. Similarly one shows
$-\sigma(\varepsilon)<\sigma(g)$. Thus $\sigma(g)\prec 1$. ∎

######  Corollary 4.5.

Let $\varphi(y)$ be an $\mathcal{L}_{\operatorname{OR}}$-formula,
$f\in{{\boldsymbol{k}}}\lfloor X\rfloor^{n}$, and
$\sigma\colon{{\boldsymbol{k}}}\lfloor X\rfloor\to\Omega$ be a ring morphism
to a real closed field $\Omega$ such that $\Omega\models\varphi(\sigma(f))$.
Then $K\models\varphi(f(a))$ for some $a\in\smallo^{m}$.

###### Proof.

Let $\mathcal{O}_{\Omega}$ be the convex hull of $\sigma({{\boldsymbol{k}}})$
in $\Omega$. By Lemma 4.4, $\sigma$ is a local ring morphism
${{\boldsymbol{k}}}\lfloor X\rfloor\to\mathcal{O}_{\Omega}$. If
$\Omega=\mathcal{O}_{\Omega}$ then model completeness of the
$\mathcal{L}_{\operatorname{OR}}$-theory of real closed ordered fields allows
us to replace $\Omega$ by a real closed ordered field extension of $\Omega$
(such as $\Omega(\\!(t^{\mathbb{Q}})\\!)$) to arrange
$\Omega\neq\mathcal{O}_{\Omega}$; thus $\Omega$ equipped with its dominance
relation associated to $\mathcal{O}_{\Omega}$ is a model of
$\operatorname{RCVF}$. Now the corollary follows from Proposition 4.2. ∎

We now quickly deduce Risler’s analytic version of Hilbert’s 17th Problem:

######  Corollary 4.6.

Let $f\in{{\boldsymbol{k}}}\lfloor X\rfloor$. Then $f(a)\geqslant 0$ for all
$a\in\smallo^{m}$ iff there are a non-zero $g\in{{\boldsymbol{k}}}\lfloor
X\rfloor$ and $h_{1},\dots,h_{k}\in{{\boldsymbol{k}}}\lfloor X\rfloor$ (for
some $k$) such that $fg^{2}=h_{1}^{2}+\cdots+h_{k}^{2}$.

###### Proof.

Suppose there are no such $g$ and $h_{j}$. This yields an ordering $\leqslant$
on $F:=\operatorname{Frac}({{\boldsymbol{k}}}\lfloor X\rfloor)$ such that
$f<0$. (See, e.g., [9, Corollary 1.1.11].) Let $\Omega$ be the real closure of
the ordered field $(F,{\leqslant})$ and $\sigma\colon{{\boldsymbol{k}}}\lfloor
X\rfloor\to\Omega$ be the natural inclusion, and apply Corollary 4.5 to the
$\mathcal{L}_{\operatorname{OR}}$-formula $y<0$ (where $y$ is a single
variable). This shows the “only if” direction; for “if” use Lemma 2.18. ∎

We can now give a general version of Risler’s real Nullstellensatz analogous
to Theorem 3.5. For this, we need to recall some definitions and basic facts
from real algebra. Let $I$ be an ideal of a ring $R$. One says that $I$ is
real if for every sequence $a_{1},\dots,a_{k}$ of elements of $R$:

$a_{1}^{2}+\cdots+a_{k}^{2}\in I\quad\Rightarrow\quad a_{1},\dots,a_{k}\in I.$

We have (cf. [9, Lemmas 4.1.5, 4.1.6]):

  1. (1)

If $I$ is real, then $I$ is radical;

  2. (2)

if $I$ is real and $R$ is noetherian, then all minimal prime ideals of $R$
containing $I$ are real; and

  3. (3)

if $I$ is a prime ideal, then $I$ is real iff $\operatorname{Frac}(R/I)$ is
orderable (formally real).

Next we define

$\sqrt[\operatorname{r}]{I}:=\big{\\{}a\in
R:\text{$a^{2k}+b_{1}^{2}+\cdots+b_{l}^{2}\in I$ for some $k\geqslant 1$ and
$b_{1},\dots,b_{l}\in R$}\big{\\}}.$

Then $\sqrt[\operatorname{r}]{I}$ is the smallest real ideal of $R$ containing
$I$, called the real radical of $I$. For $R={{\boldsymbol{k}}}\lfloor
X\rfloor$, we clearly have
$\operatorname{I}\\!\big{(}\\!\operatorname{Z}_{K}(I)\big{)}=\operatorname{I}\\!\big{(}\\!\operatorname{Z}_{K}(\sqrt[\operatorname{r}]{I})\big{)}$.

######  Theorem 4.7 (Risler’s Nullstellensatz).

Let $I$ be an ideal of ${{\boldsymbol{k}}}\lfloor X\rfloor$. Then

$\operatorname{I}\\!\big{(}\\!\operatorname{Z}_{K}(I)\big{)}=\sqrt[\operatorname{r}]{I}.$

###### Proof.

If we have some $a\in\operatorname{Z}_{K}(I)$, then some real prime ideal of
${{\boldsymbol{k}}}\lfloor X\rfloor$ contains $I$, namely the kernel of the
ring morphism $f\mapsto f(a)\colon{{\boldsymbol{k}}}\lfloor X\rfloor\to K$.
Hence we may assume that some real prime ideal of ${{\boldsymbol{k}}}\lfloor
X\rfloor$ contains $I$ and thus $\sqrt[\operatorname{r}]{I}$. Then there are
real prime ideals $P_{1},\dots,P_{k}$ ($k\geqslant 1$) of
${{\boldsymbol{k}}}\lfloor X\rfloor$ such that
$\sqrt[\operatorname{r}]{I}=P_{1}\cap\cdots\cap P_{k}$. We can arrange that
$I$ itself is a real prime ideal, and need to show that then
$\operatorname{I}\\!\big{(}\\!\operatorname{Z}_{K}(I)\big{)}=I$. Equip the
fraction field of the integral domain $R:={{\boldsymbol{k}}}\lfloor
X\rfloor/I$ with some field ordering, let $\Omega$ be its real closure, and
let $\sigma\colon{{\boldsymbol{k}}}\lfloor X\rfloor\to\Omega$ be the
composition of $f\mapsto f+I\colon{{\boldsymbol{k}}}\lfloor X\rfloor\to R$
with the natural inclusions $R\subseteq\Omega$. Arguing as in the proof of
Theorem 3.5, using Corollary 4.5 instead of Corollary 3.4, yields
$\operatorname{I}\\!\big{(}\\!\operatorname{Z}_{K}(I)\big{)}\subseteq I$. ∎

If ${{\boldsymbol{k}}}$ is real closed, then so is
${{\boldsymbol{k}}}(\\!(t^{*})\\!)$; thus the valued field
${{\boldsymbol{k}}}(\\!(t^{*})\\!)$ is a model of $\operatorname{RCVF}$. By
the previous theorem we hence immediately obtain, in a similar way as Theorem
3.5 implied Corollary 3.7:

######  Corollary 4.8.

Suppose ${{\boldsymbol{k}}}$ is real closed. Let
$f,g_{1},\dots,g_{n}\in{{\boldsymbol{k}}}[[X]]$ with ${f(a)=0}$ for all $a\in
t{{\boldsymbol{k}}}[[t]]^{m}$ such that $g_{1}(a)=\cdots=g_{n}(a)=0$. Then
there are $k\geqslant 1$ and
$b_{1},\dots,b_{l},h_{1},\dots,h_{n}\in{{\boldsymbol{k}}}[[X]]$ such that

$f^{2k}+b_{1}^{2}+\cdots+b_{l}^{2}=g_{1}h_{1}+\cdots+g_{n}h_{n}.$

######  Remark.

If in the context of the previous corollary $f$ and each $g_{j}$ are in
${{\boldsymbol{k}}}_{0}[[X]]$, where ${{\boldsymbol{k}}}_{0}$ is a euclidean
subfield of ${{\boldsymbol{k}}}$, then we can take the $b_{i}$, $h_{j}$ in
${{\boldsymbol{k}}}_{0}[[X]]$. To see this apply Theorem 4.7 to the
Weierstrass system $({{\boldsymbol{k}}}_{0}[[X]])$ instead of the Weierstrass
system $({{\boldsymbol{k}}}[[X]])$. Similarly with
${{\boldsymbol{k}}}_{0}[[X]]^{\operatorname{a}}$ or
${{\boldsymbol{k}}}_{0}[[X]]^{\operatorname{da}}$ in place of
${{\boldsymbol{k}}}_{0}[[X]]$.

The field ${\mathbb{R}}\\{\\!\\{t^{*}\\}\\!\\}$ is also real closed; hence we
obtain, similarly to Corollary 3.8:

######  Corollary 4.9.

Let $f,g_{1},\dots,g_{n}\in{\mathbb{R}}\\{X\\}$, and suppose that for each
$a\in{\mathbb{R}}\\{t\\}^{m}$ with $a(0)=0$ we have $f(a)=0$ whenever
$g_{1}(a)=\cdots=g_{n}(a)=0$. Then there are $k\geqslant 1$ and
$b_{1},\dots,b_{l},h_{1},\dots,h_{n}\in{\mathbb{R}}\\{X\\}$ such that

$f^{2k}+b_{1}^{2}+\cdots+b_{l}^{2}=g_{1}h_{1}+\cdots+g_{n}h_{n}.$

Finally, the previous corollary yields a version for germs of real analytic
functions:

######  Corollary 4.10.

Let $f,g_{1},\dots,g_{n}\colon U\to{\mathbb{R}}$ be real analytic functions,
where $U$ is an open neighborhood of $0$ in ${\mathbb{R}}^{m}$, such that for
all $a\in U$,

$g_{1}(a)=\cdots=g_{n}(a)\quad\Longrightarrow\quad f(a)=0.$

Then there are real analytic functions
$b_{1},\dots,b_{l},h_{1},\dots,h_{n}\colon V\to{\mathbb{R}}$, for some open
neighborhood $V\subseteq U$ of $0$ in ${\mathbb{R}}^{m}$, and some $k\geqslant
1$ such that

$f^{2k}+b_{1}^{2}+\cdots+b_{l}^{2}=g_{1}h_{1}+\cdots+g_{n}h_{n}\qquad\text{ on
$V$.}$

######  Remark.

In Corollary 4.10, if the germs of $f$ and of the $g_{j}$ at $0$ are algebraic
(over ${\mathbb{R}}[X]$), then we can choose the $b_{i}$, $h_{j}$ such that
their germs at $0$ are also algebraic. Similarly with “differentially
algebraic” in place of “algebraic”. (Cf. the remark following Corollary 4.8.)

##  5\. The Theory $p\operatorname{CVF}$

Our aim here is to formulate and prove a theorem analogous to Theorem 4.1 in
the $p$-adic context. We first recall the basic algebraic and model-theoretic
facts about $p$-adically closed fields. Throughout this section $K$ is a field
of characteristic zero.

###  5.1. $p$-valued fields

Recall that a valuation on $K$ with residue field ${\mathbb{F}}_{p}$ such that
$p$ is an element of smallest positive valuation is called a $p$-valuation on
$K$. The valuation ring of a $p$-valuation on $K$ is said to be a
$p$-valuation ring of $K$, and a field of characteristic zero equipped with
one of its $p$-valuation rings is a $p$-valued field. If $K$ is $p$-valued,
then so is each valued subfield of $K$. We also note:

######  Lemma 5.1.

Let $K$ be a valued field and $\Delta\neq\\{0\\}$ be a convex subgroup of its
value group. Then $K$ is $p$-valued iff the $\Delta$-specialization $\dot{K}$
of $K$ is $p$-valued, and in this case the $\Delta$-coarsening of $K$ is not
$p$-valued.

###### Proof.

The valued field $\dot{K}$ has value group $\Delta$, and its residue field is
isomorphic to $\operatorname{res}(K)$. ∎

###  5.2. $p$-adically closed fields

One says that a $p$-valued field is $p$-adically closed if it has no proper
algebraic $p$-valued field extension. The fundamental result about
$p$-adically closed fields is the following (see [48, Theorem 3.1]):

######  Theorem 5.2.

A $p$-valued field is $p$-adically closed iff it is henselian and its value
group is a ${\mathbb{Z}}$-group.

Using the Ax-Kochen/Eršov Theorem for unramified henselian valued fields of
mixed characteristic (cf., e.g., [21, Theorem 7.1]) this implies:

######  Corollary 5.3.

Any two $p$-adically closed valued fields (viewed as
$\mathcal{L}_{\preceq}$-structures as usual) are elementarily equivalent.

We also recall a well-known fact:

######  Lemma 5.4.

Let $\mathcal{O}$ be a $p$-valuation ring of $K$, and set $\varepsilon:=3$ if
$p=2$ and $\varepsilon:=2$ otherwise. Then

$\mathcal{O}\ \supseteq\ \mathcal{O}_{0}:=\big{\\{}a\in K:\text{there is some
$b\in K$ with $1+pa^{\varepsilon}=b^{\varepsilon}$}\big{\\}},$

with $\mathcal{O}=\mathcal{O}_{0}$ if the valued field $(K,\mathcal{O})$ is
henselian.

In particular, if $(K,\mathcal{O})$ is $p$-adically closed, then $\mathcal{O}$
is the unique $p$-valuation ring of $K$. Theorem 5.2 in combination with
Lemmas 1.7, 5.1, and [3, Lemma 3.4.2] also yields a specialization result for
$p$-adic closedness:

######  Lemma 5.5.

Let $K$, $\Delta$ be as in Lemma 5.1. Then $K$ is $p$-adically closed iff the
$\Delta$-coarsening of $K$ is henselian with divisible value group and the
$\Delta$-specialization $\dot{K}$ of $K$ is $p$-adically closed.

In the next lemma and its corollary we let $L\supseteq K$ be an extension of
$p$-valued fields.

######  Lemma 5.6.

If $L$ is henselian, then

$\text{$K$ is algebraically closed in
$L$}\quad\Longleftrightarrow\quad\text{$K$ is henselian and
$\Gamma_{L}/\Gamma$ is torsion-free.}$

For a proof of this lemma see [48, Lemma 4.2]. Lemma 5.6 in combination with
Lemma 1.8 and Theorem 5.2 implies:

######  Corollary 5.7.

If $L$ is $p$-adically closed, then the following are equivalent:

  1. (1)

$K$ is $p$-adically closed;

  2. (2)

$K$ is algebraically closed in $L$;

  3. (3)

$K$ is henselian and $\Gamma_{L}/\Gamma$ is torsion-free.

A field which carries at least one $p$-valuation is said to be formally
$p$-adic. Every formally $p$-adic field has characteristic zero. If $K$ is
formally $p$-adic and satisfies one of the equivalent conditions in the next
lemma, then $K$ is called $p$-adically closed.

######  Lemma 5.8.

For a formally $p$-adic field $K$, the following statements are equivalent:

  1. (1)

$K$ has no proper algebraic formally $p$-adic field extension;

  2. (2)

each $p$-valuation ring of $K$ makes $K$ a $p$-adically closed valued field;

  3. (3)

some valuation ring of $K$ makes $K$ a $p$-adically closed valued field.

###### Proof.

The implications (1) $\Rightarrow$ (2) $\Rightarrow$ (3) are clear. Let
$\mathcal{O}$ be a valuation ring of $K$ making $K$ a $p$-adically closed
valued field. Let $F$ be an algebraic field extension of $K$, and let
$\mathcal{O}_{F}$ be a $p$-valuation ring of $F$. Then $\mathcal{O}_{F}\cap K$
is a $p$-valuation ring of $K$, so $\mathcal{O}_{F}\cap K=\mathcal{O}$ and
thus $F=K$ since $(K,\mathcal{O})$ is $p$-adically closed. ∎

Below we will have occasion to discuss field equipped with two dependent
valuations, one of which is a $p$-valuation. To facilitate keeping them apart
we now generally denote $p$-valuation rings of $K$ by $\mathcal{O}_{p}$, with
maximal ideal $\smallo_{p}$, valuation $v_{p}\colon K^{\times}\to\Gamma_{p}$,
and associated dominance relation $\preceq_{p}$. If we want to make the
dependence on $K$ explicit, we write $\mathcal{O}_{K,p}$ instead of
$\mathcal{O}_{p}$, etc. (Below, $K$ will often have a unique $p$-valuation
ring, making these notations unambiguous in this case.)

######  Examples.

Let ${{\boldsymbol{k}}}$ be a $p$-adically closed field. Let
$v_{{{\boldsymbol{k}}}}\colon{{\boldsymbol{k}}}^{\times}\to\Gamma_{{{\boldsymbol{k}}}}$
be the $p$-valuation of ${{\boldsymbol{k}}}$, with valuation ring
$\mathcal{O}_{{{\boldsymbol{k}}}}$ and maximal ideal
$\smallo_{{{\boldsymbol{k}}}}$. Consider the Hahn field
$K={{\boldsymbol{k}}}(\\!(t^{\Gamma})\\!)$ where $\Gamma\neq\\{0\\}$ is
divisible. Then the $t$-adic valuation $v\colon K^{\times}\to\Gamma$ of $K$ is
not a $p$-valuation. However, the field $K$ is $p$-adically closed: Let
$\Gamma_{p}:=\Gamma\times\Gamma_{{{\boldsymbol{k}}}}$ equipped with the
lexicographic ordering, and for $f=\sum_{\gamma\in\Gamma}f_{\gamma}\gamma\in
K^{\times}$ ($f_{\gamma}\in{{\boldsymbol{k}}}$) set

$v_{p}(f):=\big{(}\delta,v_{{{\boldsymbol{k}}}}(f_{\delta})\big{)}\text{ where
$\delta=vf$;}$

then $v_{p}\colon K^{\times}\to\Gamma_{p}$ is the $p$-valuation of $K$. (Cf.
Lemma 5.5.) The valuation ring of $v_{p}$ is
$\mathcal{O}_{p}=\mathcal{O}_{{{\boldsymbol{k}}}}+\smallo$, with maximal ideal
$\smallo_{{{\boldsymbol{k}}}}+\smallo$, where $\smallo$ is the maximal ideal
of the valuation ring $\mathcal{O}={{\boldsymbol{k}}}+\smallo$ of $v$. For
$\Gamma={\mathbb{Q}}$, the $p$-valuation on
${{\boldsymbol{k}}}(\\!(t^{{\mathbb{Q}}})\\!)$ restricts to a $p$-valuation on
its subfield ${{\boldsymbol{k}}}(\\!(t^{*})\\!)$, making
${{\boldsymbol{k}}}(\\!(t^{*})\\!)$ a $p$-adically closed valued field. For
${{\boldsymbol{k}}}={\mathbb{Q}}_{p}$, the $p$-valuation on
${\mathbb{Q}}_{p}(\\!(t^{*})\\!)$ further restricts to a $p$-valuation on its
subfield ${\mathbb{Q}}_{p}\\{\\!\\{t^{*}\\}\\!\\}$, with valuation ring
${\mathbb{Z}}_{p}+\smallo_{t}$ having maximal ideal
$p{\mathbb{Z}}_{p}+\smallo_{t}$.

The subgroup $(K^{\times})^{n}={\\{a^{n}:a\in K^{\times}\\}}$ of the
multiplicative group of a $p$-adically closed field $K$ consisting of the
$n$th powers of non-zero elements of $K$ plays an important role in the study
of $K$. For $K={\mathbb{Q}}_{p}$ we recall a consequence of compactness of
${\mathbb{Z}}_{p}$ and density of ${\mathbb{Z}}$ in ${\mathbb{Z}}_{p}$:

######  Lemma 5.9.

Let $n\geqslant 1$; then
$\big{[}{\mathbb{Q}}_{p}^{\times}:({\mathbb{Q}}_{p}^{\times})^{n}\big{]}<\infty$,
and each coset of the subgroup $({\mathbb{Q}}_{p}^{\times})^{n}$ of
${\mathbb{Q}}_{p}^{\times}$ has a representative in ${\mathbb{Z}}$.

We now let $\mathcal{L}_{\operatorname{Mac}}$ be the expansion of the language
$\mathcal{L}_{\operatorname{R}}$ of rings by distinct unary predicate symbols
$\operatorname{P}_{n}$ (${n\geqslant 1}$), a binary relation symbol
$\preceq_{p}$, and a unary function symbol -1. In the following we construe a
$p$-valued field $K$ as an $\mathcal{L}_{\operatorname{Mac}}$-structure by
interpreting $\operatorname{P}_{n}$ by the set $(K^{\times})^{n}$, the binary
relation symbol $\preceq_{p}$ by the valuation of $K$, and -1 by the
multiplicative inverse function $a\mapsto a^{-1}$ extended to a total function
${K\to K}$ via $0^{-1}:=0$. Note that the class of $p$-adically closed valued
fields is $\mathcal{L}_{\operatorname{Mac}}$-axiomatizable thanks to Theorem
5.2. We let $p\operatorname{CF}$ be the
$\mathcal{L}_{\operatorname{Mac}}$-theory of $p$-adically closed fields.

######  Theorem 5.10 (Macintyre).

$p\operatorname{CF}$ admits QE.

We refer to [41] or [48, Theorem 5.6] for proofs of this theorem. For later
use we note that the symbols $\preceq_{p}$ and -1 are not actually necessary
to achieve QE. For this let $\mathcal{L}^{*}_{\operatorname{Mac}}$ be the
reduct $\mathcal{L}_{\operatorname{R}}\cup\\{\operatorname{P}_{n}:n\geqslant
1\\}$ of $\mathcal{L}_{\operatorname{Mac}}$.

######  Lemma 5.11.

Every quantifier-free $\mathcal{L}_{\operatorname{Mac}}$-formula is
$p\operatorname{CF}$-equivalent to a quantifier-free
$\mathcal{L}^{*}_{\operatorname{Mac}}$-formula.

###### Proof.

Take representatives $\lambda_{1},\dots,\lambda_{N}\in{\mathbb{Z}}$
($N\geqslant 1$) for the cosets of the subgroup
$({\mathbb{Q}}_{p}^{\times})^{n}$ of ${\mathbb{Q}}_{p}^{\times}$. Let $K$ be a
$p$-adically closed valued field. Then $\lambda_{1},\dots,\lambda_{N}$ are
also representatives for the cosets of the subgroup $(K^{\times})^{n}$ of
$K^{\times}$, by completeness of $p\operatorname{CF}$ (Corollary 5.3), hence
for $a,b\in K$, $b\neq 0$:

$a/b\in(K^{\times})^{n}\quad\Longleftrightarrow\quad
a\lambda_{i},b\lambda_{i}\in(K^{\times})^{n}\text{ for some
$i\in\\{1,\dots,N\\}$.}$

Also, let $\varepsilon$ be as in Lemma 5.4; then for $a,b\in K$ of we have

$a\preceq_{p}b\quad\Longleftrightarrow\quad\text{$b=0$, or $b\neq 0$ and
$\big{(}1+p(a/b)^{\varepsilon}\big{)}\in(K^{\times})^{\varepsilon}$.}$

This easily yields the claim. ∎

Call a field $K$ $p$-euclidean if it is formally $p$-adic and
$\big{[}K^{\times}:(K^{\times})^{n}\big{]}=\big{[}{\mathbb{Q}}_{p}^{\times}:({\mathbb{Q}}_{p}^{\times})^{n}\big{]}$
for each $n\geqslant 1$.

######  Lemma 5.12.

Suppose $K$ is $p$-euclidean. Then $K$ has a unique expansion to an
$\mathcal{L}_{\operatorname{Mac}}$-structure which is a substructure of a
model of $p\operatorname{CF}$. (In particular, $K$ has a unique $p$-valuation
ring.)

###### Proof.

Since $K$ is formally $p$-adic, it has a $p$-adically closed extension $F$,
and this gives rise to an expansion of $K$ to an
$\mathcal{L}_{\operatorname{Mac}}$-substructure of $F$ viewed as an
$\mathcal{L}_{\operatorname{Mac}}$-structure. By Lemma 5.9 and
$\big{[}K^{\times}:(K^{\times})^{n}\big{]}=\big{[}{\mathbb{Q}}_{p}^{\times}:({\mathbb{Q}}_{p}^{\times})^{n}\big{]}$
we have ${(F^{\times})^{n}\cap K^{\times}}=(K^{\times})^{n}$ for each
$n\geqslant 1$. Let $\mathcal{O}_{p}$ be the valuation ring of the
$\mathcal{L}_{\operatorname{Mac}}$-structure $K$, and let $\varepsilon$ and
$\mathcal{O}_{0}$ be as in Lemma 5.4. Then
$\mathcal{O}_{0}\subseteq\mathcal{O}_{p}$ by that lemma, and we claim
$\mathcal{O}_{p}=\mathcal{O}_{0}$. For this, let $a\in\mathcal{O}_{p}$; then
$1+pa^{\varepsilon}\in(F^{\times})^{\varepsilon}\cap
K^{\times}=(K^{\times})^{\varepsilon}$, so we get $a\in\mathcal{O}_{0}$. ∎

A field which satisfies the conclusion of the last lemma called weakly
$p$-euclidean. We always construe a weakly $p$-euclidean field as a valued
field via its unique $p$-valuation ring. Every $p$-adically closed field is
$p$-euclidean; but ${\mathbb{Q}}$ is not. However:

######  Lemma 5.13.

The field ${\mathbb{Q}}$ is weakly $p$-euclidean.

###### Proof.

Clearly ${\mathbb{Q}}$ has only one $p$-valuation ring, namely

${\mathbb{Z}}_{(p)}=\left\\{\frac{a}{b}:a,b\in{\mathbb{Z}},\ b\notin
p{\mathbb{Z}}\right\\},$

and thanks to Corollary 5.3 we also have
$({\mathbb{Q}}_{p}^{\times})^{n}\cap{\mathbb{Q}}^{\times}=(K^{\times})^{n}\cap{\mathbb{Q}}^{\times}$
for each $K\models p\operatorname{CF}$ and $n\geqslant 1$. ∎

######  Question.

Are there $p$-euclidean algebraic number fields?

###  5.3. $p$-convex valuations

Let $(K,\mathcal{O}_{p})$ be a $p$-valued field. A subring $\mathcal{O}$ of
$K$ is said to be $p$-convex if it is convex in $(K,\mathcal{O}_{p})$, that
is, if $\mathcal{O}_{p}\subseteq\mathcal{O}$. (See Section 1.4.) Every such
$p$-convex subring of $K$ is a valuation ring of $K$. If $\preceq$ is the
dominance relation on $K$ associated to a $p$-convex valuation ring of $K$,
then

$a\preceq_{p}b\ \Longrightarrow\ a\preceq b,\qquad a\prec b\ \Longrightarrow\
a\prec_{p}b\qquad\text{for all $a,b\in K$.}$

If $K\subseteq L$ is an extension of $p$-valued fields and $\mathcal{O}_{L}$
is a $p$-convex subring of $L$, then $\mathcal{O}_{L}\cap K$ is a $p$-convex
subring of $K$. In the next lemma and its corollary we let $(K,\mathcal{O})$
be an arbitrary valued field with value group $\Gamma$ and residue field
${{\boldsymbol{k}}}$.

######  Lemma 5.14.

$\text{$K$ is a $p$-adically closed field
and~{}$\mathcal{O}_{p}\subset\mathcal{O}$}\quad\Longleftrightarrow\quad\\\
\text{$(K,\mathcal{O})$ is henselian, $\Gamma$ is divisible, and
${{\boldsymbol{k}}}$ is $p$-adically closed.}$

###### Proof.

Suppose the field $K$ is $p$-adically closed and
$\mathcal{O}_{p}\subset\mathcal{O}$. Let $\Delta$ be the convex subgroup of
$\Gamma_{p}$ such that the $\Delta$-coarsening of $(K,\mathcal{O}_{p})$ is
$(K,\mathcal{O})$. Then $\Delta\neq\\{0\\}$, and by Lemma 5.5,
$(K,\mathcal{O})$ is henselian, its value group is divisible, and the
$\Delta$-specialization of $(K,\mathcal{O}_{p})$ is $p$-adically closed; the
underlying field of the latter is ${{\boldsymbol{k}}}=\mathcal{O}/\smallo$.
Conversely, suppose $(K,\mathcal{O})$ is henselian with divisible value group,
and let $\mathcal{O}_{{{\boldsymbol{k}}}}$ be a $p$-valuation ring of
${{\boldsymbol{k}}}$ making ${{\boldsymbol{k}}}$ a $p$-adically closed valued
field. Take a valuation ring $\mathcal{O}_{0}$ of $K$ contained in
$\mathcal{O}$ and a convex subgroup $\Delta_{0}$ of the value group of
$\mathcal{O}_{0}$ such that the $\Delta_{0}$-coarsening of
$(K,\mathcal{O}_{0})$ is $(K,\dot{\mathcal{O}_{0}})=(K,\mathcal{O})$ and the
$\Delta_{0}$-specialization of $(K,\mathcal{O}_{0})$ is
$({{\boldsymbol{k}}},\mathcal{O}_{{{\boldsymbol{k}}}})$. Note that
$\Delta_{0}\neq\\{0\\}$, so $\mathcal{O}_{0}\subset\mathcal{O}$. Also,
$(K,\dot{\mathcal{O}_{0}})=(K,\mathcal{O})$ is henselian, its value group
$\Gamma_{0}/\Delta_{0}\cong\Gamma$ is divisible, and the
$\Delta_{0}$-specialization of $(K,\mathcal{O}_{0})$ is $p$-adically closed;
hence $(K,\mathcal{O}_{0})$ is $p$-adically closed by Lemma 5.5. ∎

######  Corollary 5.15.

$\text{$K$ is $p$-adically closed and $\mathcal{O}$ is a non-trivial
$p$-convex subring of $K$}\ \Longleftrightarrow\ \\\ \text{$(K,\mathcal{O})$
is henselian, $\Gamma\neq\\{0\\}$ is divisible, and ${{\boldsymbol{k}}}$ is
$p$-adically closed.}$

######  Example.

Let $\Gamma\neq\\{0\\}$ be a divisible ordered abelian group, written
additively, and let ${{\boldsymbol{k}}}$ be a $p$-adically closed field. Then
the valuation ring of the $t$-adic valuation on the Hahn field
$K={{\boldsymbol{k}}}(\\!(t^{\Gamma})\\!)$ is a non-trivial $p$-convex subring
of $K$. Similarly, the $t$-adic valuation ring ${{\boldsymbol{k}}}[[t^{*}]]$
of field ${{\boldsymbol{k}}}(\\!(t^{*})\\!)$ of Puiseux series over
${{\boldsymbol{k}}}$ is a non-trivial $p$-convex subring of
${{\boldsymbol{k}}}(\\!(t^{*})\\!)$.

If $K$ is a $p$-valued field, we define the $p$-convex hull (in $K$) of a
subring $R$ of $K$ to be the convex hull of $R$ in the valued field $K$.

######  Lemma 5.16.

Let $(K,\mathcal{O}_{p})$ be a $p$-valued field and let $F$ be a subfield of
$K$, with $p$-convex hull $R$ in $K$. Then $R\neq\mathcal{O}_{p}$, hence if
some $p$-convex subring $\mathcal{O}\neq K$ of $K$ contains $F$, then $R$ is a
non-trivial $p$-convex subring of $K$.

###### Proof.

For the first statement use that $\mathcal{O}_{p}$ does not contain a subfield
of $K$; the rest follows from this. ∎

We now let $\mathcal{L}=\mathcal{L}_{\operatorname{Mac},{\preceq}}$ be the
expansion of the language $\mathcal{L}_{\operatorname{Mac}}$ of Macintyre by a
single binary relation symbol $\preceq$. Let $p\operatorname{CVF}$ be the
$\mathcal{L}$-theory expanding $p\operatorname{CF}$ by axioms which state that
$\preceq$ is interpreted by a dominance relation associated to a non-trivial
$p$-convex valuation ring.

######  Theorem 5.17.

$p\operatorname{CVF}$ has QE.

This $p$-adic analogue of Theorem 4.1 is stated without proof in [12, Theorem
5]; see also [6, Corollaire 2.3(1)] and [28, Lemma 2.13]. For the proof, these
sources quote the unpublished thesis [15] or the notationally dense and long
paper [63] (which also yields a primitive recursive algorithm for QE in
$p\operatorname{CVF}$). For the convenience of the reader we include a
deduction of the important Theorem 5.17 below from standard facts in the
literature. We start with a somewhat weaker result:

######  Proposition 5.18.

$p\operatorname{CVF}$ is model complete.

This follows from the Ax-Kochen/Eršov Theorem for elementary substructure:

######  Theorem 5.19.

Let $K\subseteq E$ be an extension of valued fields. Then

$K\preceq
E\quad\Longleftrightarrow\quad\text{$\operatorname{res}(K)\preceq\operatorname{res}(E)$
and $\Gamma\preceq\Gamma_{E}$.}$

For this in a $3$-sorted context, for valued fields equipped with an angular
component map, see [21, Corollary 5.23]. One may reduce to this setting by
replacing $(E,K)$ by an $\aleph_{1}$-saturated elementary extension, after
which $E$ can be equipped with a cross-section which restricts to a cross-
section of $K$ (cf. [3, Lemmas 3.3.39, 3.3.40]).

###### Proof of Proposition 5.18.

Let $K\subseteq E$ be models of $p\operatorname{CVF}$. By Corollary 5.15, the
valued field $E$ is henselian, $\Gamma_{E}\neq\\{0\\}$ is divisible, and
$\operatorname{res}(E)$ is $p$-adically closed; similarly for $K$, $\Gamma$,
$\operatorname{res}(K)$ in place of $E$, $\Gamma_{E}$,
$\operatorname{res}(E)$, respectively. We have
$\operatorname{res}(K)\preceq\operatorname{res}(E)$ by Theorem 5.10, and also
$\Gamma\preceq\Gamma_{E}$ (see, e.g., [3, Example B.11.12]). Hence $K$ is an
elementary $\mathcal{L}_{\preceq}$-substructure of $E$, by Theorem 5.19. By
Corollary 5.7, $K$ is algebraically closed in $E$, so $(E^{\times})^{n}\cap
K=(K^{\times})^{n}$ for each $n\geqslant 1$. Hence also using Lemma 5.4 we
obtain that $K$ is also an elementary $\mathcal{L}$-substructure of $E$. ∎

By Proposition 5.18 we have

${\mathbb{Q}}_{p}\\{\\!\\{t^{*}\\}\\!\\}\ \preceq\
{\mathbb{Q}}_{p}(\\!(t^{*})\\!)\ \preceq\ \
\operatorname{cl}({\mathbb{Q}}_{p}[t^{{\mathbb{Q}}}])\ \preceq\
{\mathbb{Q}}_{p}(\\!(t^{{\mathbb{Q}}})\\!)$

as $\mathcal{L}$-structures. In order to finish the proof of Theorem 5.17, we
recall a well-known fact about $n$th powers in henselian valued fields of
equicharacteristic zero:

######  Lemma 5.20.

Let $K$ be a henselian valued field of equicharacteristic zero, and suppose
$n\geqslant 1$. Then

$(\mathcal{O}^{\times})^{n}=(K^{\times})^{n}\cap\mathcal{O}^{\times}=\big{\\{}a\in\mathcal{O}^{\times}:\overline{a}\in({{\boldsymbol{k}}}^{\times})^{n}\big{\\}}.$

Hence for $a,b\in K^{\times}$ with $a\sim b$ we have
$a\in(K^{\times})^{n}\Leftrightarrow b\in(K^{\times})^{n}$.

###### Proof of Theorem 5.17.

Let $E,F\models p\operatorname{CVF}$ and let $K$ be a common substructure of
$E$ and of $F$; we need to show $E_{K}\equiv F_{K}$; cf. [3, Corollary
B.11.6]. The valuation ring $\mathcal{O}$ of $K$ is $p$-convex. Note that
$\operatorname{res}(K)=\mathcal{O}/\smallo\subseteq\operatorname{res}(E)$ has
characteristic zero, thus $\mathcal{O}_{p}\subset\mathcal{O}$. We reduce to
the case $\Gamma\neq\\{0\\}$. Suppose $\Gamma=\\{0\\}$. Replacing $E$, $F$ by
elementary extensions we first arrange that $E$, $F$ are
$\aleph_{1}$-saturated, hence we have cross-sections $s_{E}\colon\Gamma_{E}\to
E^{\times}$ of $E$ and $s_{F}\colon\Gamma_{F}\to F^{\times}$ of $F$. Take any
$\alpha\in\Gamma_{E}^{>}$ and set $x:=s_{E}(\alpha)\in E^{\times}$; then $x$
is transcendental over $K$ by [3, Corollary 3.1.8]. Since $\Gamma_{E}$ is
divisible, we have $x\in(E^{\times})^{n}$ for each $n\geqslant 1$. Similarly
we take any $\beta\in\Gamma_{F}^{>}$ and set $y:=s_{F}(\beta)\in F^{\times}$;
then $y$ is transcendental over $K$ and $y\in(F^{\times})^{n}$ for each
$n\geqslant 1$. The field isomorphism $\sigma\colon K(x)\to K(y)$ over $K$
with $\sigma(x)=y$ is a valued field isomorphism [3, Lemma 3.1.30]. Let $0\neq
f\in K[T]$, where $T$ is a single variable over $K$, and $n\geqslant 1$. Let
$a\in K^{\times}$, $g\in K[T]$, and $m$ be such that $f=aT^{m}(1+gT)$. Then
$f(x)\sim ax^{m}$, hence by Lemma 5.20:

$f(x)\in(E^{\times})^{n}\quad\Longleftrightarrow\quad
ax^{m}\in(E^{\times})^{n}\quad\Longleftrightarrow\quad a\in(E^{\times})^{n}.$

Similarly we obtain $f(y)\in(F^{\times})^{n}\Leftrightarrow
a\in(F^{\times})^{n}$. Since $K$ is an
$\mathcal{L}_{\operatorname{Mac}}$-substructure of both $E$ and $F$, this
yields $f(x)\in(E^{\times})^{n}\Leftrightarrow f(y)\in(F^{\times})^{n}$. If
also $0\neq g\in K[T]$ and $h:=f/g\in K(T)$, then

$h(x)\in(E^{\times})^{n}\ \Longleftrightarrow\
(fg^{n-1})(x)/g(x)^{n}\in(E^{\times})^{n}\ \Longleftrightarrow\
(fg^{n-1})(x)\in(E^{\times})^{n},$

and similarly with $F$ and $y$ in place of $E$ and $x$, respectively. Hence by
the above applied to $fg^{n-1}\in K[T]$ in place of $f$ we obtain
$h(x)\in(E^{\times})^{n}\Leftrightarrow h(y)\in(F^{\times})^{n}$. Using Lemma
5.4, this also implies $h(x)\preceq_{p}1\Leftrightarrow h(y)\preceq_{p}1$.
Hence $\sigma$ is an isomorphism $K(x)\to K(y)$ between
$\mathcal{L}_{\operatorname{Mac}}$-substructures of $E$ and of $F$,
respectively. Now identify $K(x)$ with its image under $\sigma$ and replace
$K$ by $K(x)$ to arrange $\Gamma\neq\\{0\\}$.

Let $L$ be the algebraic closure of $K$ inside $E$. Then $L$ is $p$-adically
closed, by Corollary 5.7. Now [48, Theorem 3.10] yields a $K$-embedding $j$ of
the $p$-valued field $L$ into $F$, and $j$ is also an embedding of
$\mathcal{L}_{\operatorname{Mac}}$-structures. By Lemma 1.6, there is a unique
$p$-convex subring of $L$ lying over $\mathcal{O}$, hence $j$ is also an
embedding of $\mathcal{L}$-structures. Identifying $L$ with its image under
$j$ we first arrange that $L$ is a common $\mathcal{L}$-substructure of $E$
and $F$, and replacing $K$ by $L$ we then arrange that $K\models
p\operatorname{CVF}$. Then $K\preceq E$ and $K\preceq F$ by Proposition 5.18
and thus $E\equiv_{K}F$. ∎

Let $\mathcal{L}^{*}:=\mathcal{L}^{*}_{\operatorname{Mac}}\cup\\{\preceq\\}$
be the expansion of the language of rings $\mathcal{L}_{\operatorname{R}}$ by
the unary relation symbols $\operatorname{P}_{n}$ ($n\geqslant 1$) and the
binary relation symbol $\preceq$ (a reduct of $\mathcal{L}$).

######  Corollary 5.21.

Each $\mathcal{L}$-formula is $p\operatorname{CVF}$-equivalent to a
quantifier-free $\mathcal{L}^{*}$-formula.

This follows from Lemma 5.11 and Theorem 5.17.

######  Corollary 5.22.

$p\operatorname{CVF}$ is complete: $K\equiv{\mathbb{Q}}_{p}(\\!(t^{*})\\!)$
for each $K\models p\operatorname{CVF}$.

###### Proof.

Expand the field ${\mathbb{Q}}$ to an $\mathcal{L}$-structure by interpreting
$\preceq_{p}$ by the $p$-adic dominance relation on ${\mathbb{Q}}$, $\preceq$
by the trivial dominance relation on ${\mathbb{Q}}$, and
$\operatorname{P}_{n}$ by
$({\mathbb{Q}}_{p}^{\times})^{n}\cap{\mathbb{Q}}^{\times}$ ($n\geqslant 1$).
Note that
$({\mathbb{Q}}_{p}^{\times})^{n}\cap{\mathbb{Q}}=(K^{\times})^{n}\cap{\mathbb{Q}}$
for each $K\models p\operatorname{CF}$ and $n\geqslant 1$, thanks to Corollary
5.3. This $\mathcal{L}$-structure embeds into each model of
$p\operatorname{CVF}$ (see Lemma 5.13), so the corollary follows from Theorem
5.17. ∎

In [5], Bélair gives an explicit axiomatization of the universal part
$p\operatorname{CF}_{\forall}$ of the
$\mathcal{L}_{\operatorname{Mac}}$-theory $p\operatorname{CF}$. We note:

######  Corollary 5.23.

The $\mathcal{L}$-theory $p\operatorname{CVF}_{\forall}$ is axiomatized by

$p\operatorname{CF}_{\forall}\cup\big{\\{}1\preceq n\cdot 1:n\geqslant
1\big{\\}}\cup\big{\\{}\forall x\forall y(x\preceq_{p}y\rightarrow x\preceq
y)\big{\\}}.$

###### Proof.

Clearly $p\operatorname{CVF}_{\forall}$ contains the displayed
$\mathcal{L}$-theory. Conversely, let $K$ be a model of the latter. Then the
valuation ring $\mathcal{O}$ associated to the dominance relation $\preceq$ on
$K$ is $p$-convex, and its residue field has characteristic zero. Take an
extension $K_{1}$ of the $\mathcal{L}_{\operatorname{Mac}}$-reduct of $K$ to a
model of $p\operatorname{CF}$, and expand it to an $\mathcal{L}$-structure by
interpreting $\preceq$ by the dominance relation associated to the $p$-convex
hull of $\mathcal{O}$ in $K_{1}$. Then $K\subseteq K_{1}$ as
$\mathcal{L}$-structures, so replacing $K$ by $K_{1}$ we can arrange that
$K\models p\operatorname{CF}$. If the $p$-convex subring $\mathcal{O}$ of $K$
is non-trivial, then we are done. Suppose otherwise; then $\mathcal{O}=K$.
Then by the example following Corollary 5.15, the valuation ring of the
$t$-adic valuation on the Hahn field $L:=K(\\!(t^{\mathbb{Q}})\\!)$ is a non-
trivial $p$-convex subring of $L$, and accordingly expanding $L$ to an
$\mathcal{L}$-structure yields an extension of $K$ to a model of
$p\operatorname{CVF}$ as required. ∎

##  6\. The $p$-adic Analytic Nullstellensatz

In this section we let $K$ be a field of characteristic zero. We first recall
the definition and a few basic facts about the $p$-adic Kochen operator
$\gamma=\gamma_{p}$, the $p$-adic Kochen ring $\Lambda_{K}$, and generalize
some auxiliary results about $p$-adic ideals from [59, 60]. In Proposition
6.18 we establish a specialization result analogous to Propositions 3.2 and
4.2, from which a general form of the $p$-adic analytic version of Hilbert’s
17th Problem (Corollary 6.22) and the $p$-adic analytic Nullstellensatz
(Theorem 6.24) follow. Theorems A, B, and C from the introduction are then
easy consequences. (See Corollaries 6.25–6.29.)

###  6.1. The Kochen ring

We set $K_{\infty}:=K\cup\\{\infty\\}$ where $\infty\notin K$. For $f\in K$ we
put $\wp(f):=f^{p}-f$ and

$\gamma(f):=\frac{1}{p}\frac{\wp(f)}{\wp(f)^{2}-1}\in
K_{\infty}\quad\text{where $\gamma(f):=\infty$ if $\wp(f)=\pm 1$.}$

For the following see [48, Lemma 6.1]:

######  Lemma 6.1.

Let $\mathcal{O}$ be a valuation ring of $K$. Then $\mathcal{O}$ is a
$p$-valuation ring of $K$ iff $p\notin\mathcal{O}^{\times}$ and
$\gamma(f)\in\mathcal{O}$ for all $f\in K$.

By convention, for $R\subseteq K$ we set $\gamma(R):=\big{\\{}\gamma(f):f\in
R\big{\\}}\setminus\\{\infty\\}$. If $K$ is formally $p$-adic, then
$\gamma(f)\neq\infty$ for each $f\in K$, by the previous lemma, so in this
case $\gamma(R)$ indeed agrees with the image of the restriction of the map
$\gamma$ to a map $R\to K_{\infty}$. For a proof of the following see [48,
Theorem 6.4]:

######  Proposition 6.2.

Let $(K,\mathcal{O})$ be $p$-valued field and $L$ be a field extension of $K$.
Then there is a $p$-valuation ring of $L$ lying over $\mathcal{O}$ iff
$p\notin\mathcal{O}[\gamma(L)]^{\times}$.

Since $K$ is formally $p$-adic iff there is a $p$-valuation ring of $K$ which
lies over the unique $p$-valuation ring ${\mathbb{Z}}_{(p)}$ of the subfield
${\mathbb{Q}}$ of $K$, Proposition 6.2 implies:

######  Corollary 6.3.

$K$ is formally $p$-adic $\Longleftrightarrow$
$p\notin{\mathbb{Z}}_{(p)}[\gamma(K)]^{\times}$.

The $p$-adic Kochen ring of $K$ is the subring

$\Lambda_{K}:=\left\\{\frac{f}{1-pg}:f,g\in{\mathbb{Z}}[\gamma(K)],\ pg\neq
1\right\\}$

of $K$. It is easy to see that

$\Lambda_{K}=\left\\{\frac{f}{1-pg}:f,g\in{\mathbb{Z}}_{(p)}[\gamma(K)],\
pg\neq 1\right\\}.$

If $K$ is formally $p$-adic then by the previous corollary we can omit here
the condition “$pg\neq 1$”. The next result is shown in [48, Theorem 6.14]:

######  Theorem 6.4.

$\Lambda_{K}=\bigcap\big{\\{}\mathcal{O}:\text{$\mathcal{O}$ is a
$p$-valuation ring of $K$}\big{\\}}$.

By Theorem 6.4 and the proof of Lemma 5.12, with $\mathcal{O}_{0}$ as in Lemma
5.4, we have $\mathcal{O}_{0}=\Lambda_{K}$ whenever $K$ is $p$-euclidean.
Also, $K$ is formally $p$-adic iff $\Lambda_{K}\neq K$; moreover:

######  Corollary 6.5.

$K$ is formally $p$-adic $\Longleftrightarrow$ $p\notin\Lambda_{K}^{\times}$.

###### Proof.

Suppose $K$ is formally $p$-adic, and towards a contradiction assume
$p\in\Lambda_{K}^{\times}$. Take $f,g\in{\mathbb{Z}}[\gamma(K)]$ with
$p^{-1}=f/(1-pg)$; then $p^{-1}=f+g\in{\mathbb{Z}}[\gamma(K)]$, contradicting
Corollary 6.3. ∎

Theorem 6.4 implies that $\Lambda_{K}$ is a Prüfer domain (every localization
of $\Lambda_{K}$ at a maximal ideal of $\Lambda_{K}$ is a valuation ring of
$K$); see [48, Corollary 6.16]. In fact, $\Lambda_{K}$ is even a Bézout domain
(every finitely generated ideal of $\Lambda_{K}$ is principal); cf. [48,
Theorem 6.17]. Hence every subring of $K$ containing $\Lambda_{K}$ is also a
Bézout domain. In particular, given a subring $R$ of $K$, the subring

$\Lambda_{K}R:=\left\\{\frac{f}{1-pg}:f\in R[\gamma(K)],\
g\in{\mathbb{Z}}[\gamma(K)],\ pg\neq 1\right\\}.$

of $K$ generated by $\Lambda_{K}$ and $R$ is a Bézout domain. (We will not use
this fact below.)

###  6.2. $p$-adic ideals and $p$-adic radical

In this subsection we let $R$ be an integral domain with fraction field $K$.
We say that an ideal $I$ of $R$ is $p$-adic if

$I=\sqrt{I\,\Lambda_{K}R}\cap R,$

that is, if $I$ contains each $r\in R$ such that $r^{k}\in I\,\Lambda_{K}R$
for some $k\geqslant 1$. (This notion was introduced in [59] and further
studied in [23, 60].) If $I$ is $p$-adic, then $I$ is radical. The trivial
ideals $I=\\{0\\}$ and $I=R$ of $R$ clearly are $p$-adic. This notion of
$p$-adic ideal is mainly of interest if $K$ is formally $p$-adic: otherwise we
have $\Lambda_{K}=K$, hence $R$ has no non-trivial $p$-adic ideals.

######  Lemma 6.6.

Let $R^{*}$ be an integral domain extending $R$ such that $R^{*}$ is
faithfully flat over $R$, and let $I$ be an ideal of $R$ such that the ideal
$IR^{*}$ of $R^{*}$ is $p$-adic. Then $I$ is $p$-adic.

###### Proof.

Let $f\in\sqrt{I\,\Lambda_{K}R}\cap R$. This yields $k\geqslant 1$ with
$f^{k}\in I\,\Lambda_{K}R$. Now with $K^{*}:=\operatorname{Frac}(R^{*})$ we
have $\Lambda_{K}\subseteq\Lambda_{K^{*}}$, hence $f^{k}\in
I\,\Lambda_{K^{*}}R^{*}$. Since $IR^{*}$ is $p$-adic we get $f\in IR^{*}$ and
thus $f\in I$ because $R^{*}$ is faithfully flat over $R$ [42, Theorem
7.5(ii)]. ∎

######  Lemma 6.7.

If $R$ is noetherian, then all minimal prime ideals of $R$ containing a given
$p$-adic ideal of $R$ are $p$-adic.

This follows from an easy ring-theoretic lemma, applied to $A=R$,
$B=\Lambda_{K}R$:

######  Lemma 6.8.

Let $A\subseteq B$ be a ring extension where $A$ is noetherian. Let $I$ be an
ideal of $A$ such that $I=\sqrt{IB}\cap A$. Then for each minimal prime ideal
$P$ of $A$ containing $I$ we also have $P=\sqrt{PB}\cap A$.

In the next lemma and its corollary we assume that $R$ is local with maximal