# Valuation rings as limits of complete intersection rings

Dorin Popescu  Simion Stoilow Institute of Mathematics of the Romanian
Academy, Research unit 5, University of Bucharest, P.O. Box 1-764, Bucharest
014700, Romania, Email<EMAIL_ADDRESS>
###### Abstract.

We show that a valuation ring containing its residue field of characteristic
$p>0$ is a filtered direct limit of complete intersection ${\bf
F}_{p}$-algebras.

## Introduction

In the idea of Zariski’s Uniformization Theorem [24], a recent result of B.
Antieau, R. Datta [2, Theorem 4.1.1] says that every perfect valuation ring of
characteristic $p>0$ is a filtered union of its smooth ${\bf
F}_{p}$-subalgebras. This result is an application of [20, Theorem 1.2.5]
which relies on some results from [6]. Also E. Elmanto and M. Hoyois proved
that an absolute integrally closed valuation ring of residue field of
characteristic $p>0$ is a filtered union of its regular finitely generated
${\bf Z}$-subalgebras (see [2, Corollary 4.2.4]). We remind that a filtered
direct limit (in other words a filtered colimit) is a limit indexed by a small
category that is filtered (see [19, 002V] or [19, 04AX]). A filtered union is
a filtered direct limit in which all objects are subobjects of the final
colimit, so that in particular all the transition arrows are monomorphisms.

It is well known that if the fraction field extension of an immediate
extension $V\subset V^{\prime}$ is finite and $p>0$ then $V^{\prime}$ could be
not a filtered direct limit of smooth $V$-algebras as shows [14, Example 3.13]
inspired from [12] (see also [14, Remark 6.10]). After seeing [8, Theorem 6.2]
(see also [21], [22]) we understood that in general we should expect that
$V^{\prime}$ is a filtered union of its complete intersection $V$-subalgebras.
This is done in Theorem 16.

A complete intersection $V$-algebra is a local $V$-algebra of type $C/(P)$,
where $C$ is a localization of a polynomial $V$-algebra and $P$ is a regular
system of elements of $C$. Thus the above theorem says that $V^{\prime}$ is a
filtered union of its $V$-subalgebras of type $C/(P)$. Since $V^{\prime}$ is
local it is enough to say that $V^{\prime}$ is a filtered union of its
$V$-subalgebras of type $T_{h}/(P)$, $T$ being a polynomial $V$-algebra,
$0\not=h\in T$ and $P$ is a regular system of elements of $T$. Clearly,
$T_{h}$ is a smooth $V$-algebra and in fact it is enough to say that
$V^{\prime}$ is a filtered union of its $V$-subalgebras of type $G/(P)$, where
$G$ is a smooth $V$-algebra and $P$ is a regular system of elements of $G$.
Conversely, a $V$-algebra of such type $G/(P)$ has the form $T_{h}/(P)$ for
some $T,h,P$ using [18, Theorem 2.5]. By abuse we understand by a complete
intersection  $V$-algebra a $V$-algebra of such type $G/(P)$, or $T_{h}/(P)$
which are not assumed to be flat over $V$. More general, we understand by a
complete intersection  algebra over a ring $A$ an $A$-algebra of type $G/(P)$,
or $T_{h}/(P)$, where $G$ is a smooth $A$-algebra, $T$ is a polynomial
$A$-algebra, $0\not=h\in T$ and $P$ is a regular system of elements of $G$,
respectively $T$ (we do not assume $G/(P)$, or $T_{h}/(P)$ flat over $A$).

The goal of this paper is to establish the following theorem.

######  Theorem 1.

Let $V$ be a valuation ring containing a perfect field $F$ of characteristic
$p>0$, $k$ its residue field, $\Gamma$ its value group and $K$ its fraction
field. Then the following statements hold

  1. (1)

if $\Gamma$ is finitely generated, $k\subset V$ then $V$ is a filtered union
of its complete intersection $F$-subalgebras,

  2. (2)

if $\Gamma$ is finitely generated, $k/F$ is separably generated (that is there
exists a transcendental basis $x$ of $k$ over $F$ such that the field
extension $F(x)\subset k$ is algebraic separable) and $V$ is Henselian, then
$V$ is a filtered union of its complete intersection $F$-subalgebras,

  3. (3)

if $k\subset V$ then $V$ is a filtered direct limit of complete intersection
$F$-algebras,

  4. (4)

if $k/F$ is separably generated and either $\dim V=1$, or $V$ is Henselian if
$\dim V>1$, then $V$ is a filtered direct limit of complete intersection
$F$-algebras.

The proof is given in Corollaries 17, 18 and Theorems 23, 27. It is possible
that the above theorem could give a complete intersection version of the local
uniformization.

For the proof we assume that the residue field $k\subset V$, the value group
$\Gamma$ finitely generated and then it is enough to study the immediate
extension $V_{0}=V\cap k(y)\subset V$, where $y$ is a set of elements from $V$
such that $\mathrm{val}(y)$ is a basis in $\Gamma$. Proposition 14 allows to
reduce to the case when $K=k(y,x)$ for an element $x\in V$ which is
transcendental over $k(y)$. By [7, Theorem 1] $x$ is a pseudo limit of a
pseudo convergent sequence $a$ over $V_{0}$ which has no pseudo limit in
$k(y)$. If $a$ is a transcendental pseudo convergent sequence then $V$ is a
filtered union of its localizations of polynomial $V_{0}$-subalgebras in one
variable by [13, Lemma 3.2] (see also [16, Lemma 15]), which cannot be applied
when $a$ is algebraic. But $V$ is a filtered union of its complete
intersection $V_{0}$-subalgebras using Theorem 11.

When $\Gamma$ is not finitely generated we had to use the cross-sections as in
[23, Lemma 7.9], or [3, 3.3.39, 3.3.40] obtained using some ultrapower methods
as we did it in [16] when $V\supset{\bf Q}$. The same methods are necessary to
reduce to the case when the residue field $k\subset V$ in the case when $V$ is
Henselian (see Theorem 25).

We owe thanks to F. V. Kuhlmann who hinted some mistakes in a previous version
and for some explanations on Theorem 22. Also we owe thanks the referee who
pointed a mistake in a previous form of Lemma 10.

##  1\. Pure transcendental immediate extensions of valuation rings

An inclusion $V\subset V^{\prime}$ of valuation rings is an _immediate
extension_ if it is local as a map of local rings and induces isomorphisms
between the value groups and the residue fields of $V$ and $V^{\prime}$.

Let $\lambda$ be a fixed limit ordinal and $v=\\{v_{i}\\}_{i<\lambda}$ a
sequence of elements in $V$ indexed by the ordinals $i$ less than $\lambda$.
Then $v$ is _pseudo convergent_ if

$\mathrm{val}(v_{i}-v_{i^{\prime\prime}})<\mathrm{val}(v_{i^{\prime}}-v_{i^{\prime\prime}})\
\ \mbox{for}\ \ i<i^{\prime}<i^{\prime\prime}<\lambda$ (see [7], [17]). A
_pseudo limit_ of $v$ is an element $v\in V$ with

$\mathrm{val}(v-v_{i})<\mathrm{val}(v-v_{i^{\prime}})\ \ \mbox{(that is,}\ \
\mathrm{val}(v-v_{i})=\mathrm{val}(v_{i}-v_{i^{\prime}}))\ \ \mbox{for}\ \
i<i^{\prime}<\lambda$. We say that $v$ is

  1. (1)

_algebraic_ if some $f\in V[T]$ satisfies
$\mathrm{val}(f(v_{i}))<\mathrm{val}(f(v_{i^{\prime}}))$ for large enough
$i<i^{\prime}<\lambda$;

  2. (2)

_transcendental_ if each $f\in V[T]$ satisfies
$\mathrm{val}(f(v_{i}))=\mathrm{val}(f(v_{i^{\prime}}))$ for large enough
$i<i^{\prime}<\lambda$.

When for any $\gamma\in\Gamma$ it holds
$\mathrm{val}(v_{i}-v_{i^{\prime}})>\gamma$ for $i<i^{\prime}<\lambda$ large
we call $v$ fundamental.

######  Lemma 2.

Let $V\subset V^{\prime}$ be an immediate extension of valuation rings,
$K\subset K^{\prime}$ its fraction field extension and $(v_{i})_{i<\lambda}$
an algebraic pseudo convergent sequence in $V$, which has a pseudo limit $x$
in $V^{\prime}$, but no pseudo limit in $K$. Set
$x_{i}=(x-v_{i})/(v_{i+1}-v_{i})$. Let $s\in{\bf N}$ be the minimal degree of
the polynomials $f\in V[Y]$ such that
$\mathrm{val}(f(v_{i}))<\mathrm{val}(f(v_{j}))$ for large $i<j<\lambda$ and
$g\in V[Y]$ a polynomial with $\operatorname{deg}g<s$. Then there exist $d\in
V\setminus\\{0\\}$ and $u\in V[x_{i}]$ for some $i<\lambda$ with $g(x)=du$ and
$\mathrm{val}(u)=0$.

###### Proof.

In the Taylor expansion111The polynomials $D^{(n)}f\in R[Y]$ for $f\in R[Y]$,
the so called Hasse-Schmidt derivatives, make sense for any ring $R$: indeed,
one constructs the Taylor expansion in the universal case $R={\bf
Z}[a_{0},\dots,a_{\operatorname{deg}f}]$ by using the equality
$n!\cdot(D^{(n)}f)=f^{(n)}$ and verifying over $\operatorname{Frac}(R)$.

$\textstyle
g(x)=\sum_{n=0}^{\operatorname{deg}g}(D^{(n)}g)(v_{i})\cdot(x-v_{i})^{n}\quad\text{with}\quad
D^{(n)}g\in V[Y]$

the values $\mathrm{val}((D^{(n)}g)(v_{i})\cdot(x-v_{i})^{n})$ are pairwise
distinct for all $n\leq\operatorname{deg}g$ and every large enough $i$ because
$\mathrm{val}(x-v_{i})=\mathrm{val}(v_{i+1}-v_{i})$ increases,
$\operatorname{deg}(D^{(n)}g)<s$ and so $\mathrm{val}((D^{(n)}g)(v_{i}))$ is
constant for all $n\leq\operatorname{deg}g$ and $i$ large. We cannot have
$\mathrm{val}((D^{(n)}g)(v_{i})\cdot(x-v_{i})^{n})<\mathrm{val}(g(v_{i}))$ for
some $n>0$ and $i$ large because otherwise we get from above that
$\mathrm{val}(g(x))=\mathrm{val}((D^{(n)}g)(v_{i})\cdot(x-v_{i})^{n})$
increases, which is false. It follows that
$\mathrm{val}((D^{(n)}g)(v_{i})\cdot(x-v_{i})^{n})>\mathrm{val}(g(v_{i}))$ for
all $n>0$ and $i$ large and so

$g(x)=g(v_{i})+\sum_{n=1}^{\operatorname{deg}g}(D^{(n)}g)(v_{i})(v_{i+1}-v_{i})^{n}x_{i}^{n}\in
g(v_{i})(1+{\mathfrak{m}}^{\prime}\cap V[x_{i}]),$

which is enough.

######  Lemma 3.

Let $V\subset V^{\prime}$ be an immediate extension of valuation rings,
$K\subset K^{\prime}$ its fraction field extension and $(v_{i})_{i<\lambda}$
an algebraic pseudo convergent sequence in $K$, which has a pseudo limit $x$
in $V^{\prime}$, algebraic over $V$, but no pseudo limit in $K$. Assume that
$h=$Irr$(x,K)$ is from $V[X]$. Then

  1. (1)

$\mathrm{val}(h(v_{i}))<\mathrm{val}(h(v_{j}))$ for large $i<j<\lambda$,

  2. (2)

if $h$ has minimal degree among the polynomials $f\in V[X]$ such that
$\mathrm{val}(f(v_{i}))<\mathrm{val}(f(v_{j}))$ for large $i<j<\lambda$, then
$V^{\prime\prime}=V^{\prime}\cap K(x)$ is a filtered union of its complete
intersection $V$-subalgebras.

###### Proof.

The first part follows from [10, Corollary 5.5]. Now, assume that
$\operatorname{deg}h$ is minimal among $f$ having the property from (2). As in
Lemma 2 we consider $(x_{i})_{i<\lambda}$ and note that for a polynomial $g\in
V[X]$ of degree $<\operatorname{deg}h$ we have $g(x)=du$ for some $d\in
V\subset\\{0\\}$ and $u\in V[x_{i}]$ for some $i<\lambda$ with
$\mathrm{val}(u)=0$. An element of $V^{\prime\prime}$ has the form $g(x)/t$
for some $g\in V[X]$ with $\operatorname{deg}g<\operatorname{deg}h$ and $t\in
V^{*}$ such that $\mathrm{val}(g(x))\geq\mathrm{val}(t)$. It follows that
$g(x)/t=(d/t)u\in V[x_{i}]_{{\mathfrak{m}}^{\prime}\cap V[x_{i}]}$, where
${\mathfrak{m}}^{\prime}$ is the maximal ideal of $V^{\prime}$.

Thus $V^{\prime\prime}$ is the union of
$(V[x_{i}]_{{\mathfrak{m}}^{\prime}\cap V[x_{i}]})_{i<\lambda}$, which is
filtered increasing as in the proof of [13, Lemma 3.2], or [16, Lemma 15]. We
have $V[x_{i}]\cong V[X_{i}]/(h_{i})$, where $h_{i}$ is defined below. Set
$g_{i}=h(v_{i}+(v_{i+1}-v_{i})X_{i})\in V[X_{i}]$. By construction
$x_{i}=(x-v_{i})/(v_{i+1}-v_{i})$ and so $g_{i}(x_{i})=h(x)=0$. As $h$ is
irreducible in $K[X]$ we get $g_{i}$ irreducible in $K[X]$ too because it is
obtained from $h$ by a linear transformation. So $K[x_{i}]\cong
K[X_{i}]/(g_{i})$. We have $g_{i}=uh_{i}$ for some primitive polynomial
$h_{i}$ of $V[X_{i}]$ and a nonzero constant $u$ of $K$. Then
$V^{\prime\prime}$ is a filtered union of its complete intersections
$V$-subalgebras.

######  Remark 4.

The extension $V\subset V^{\prime\prime}$ from (2) of the above lemma is
isomorphic with the one constructed in [7, Theorem 3].

The following lemma is a variant of Ostrowski ([12, page 371, IV and III], see
also [17, (II,4),Lemma 8]) and we need it next.

######  Lemma 5.

(Ostrowski) Let $\beta_{1},\ldots,\beta_{m}$ be any elements of an ordered
abelian group $G$, $\lambda$ a limit ordinal and let
$\\{\gamma_{s}\\}_{s<\lambda}$ be a well-ordered, monotone increasing set of
elements of G, without a last element. Let $t_{1},\ldots,t_{m}$, be distinct
integers. Then there exists an ordinal $\nu<\lambda$ such that
$\beta_{i}+t_{i}\gamma_{s}$ are different for all $s>\nu$. In particular,
there exists an integer $1\leq r\leq m$ such that

$\beta_{i}+t_{i}\gamma_{s}>\beta_{r}+t_{r}\gamma_{s}$

for all $i\not=r$ and $s>\nu$.

###### Proof.

The known proof from the quoted papers gives the second statement even when
the integers $t_{i}$ are not necessarily positive. Then apply this statement
iteratively to get the first statement.

Next we give a certain extension of the above lemma. By $|t|$ we denote the
absolute value of the integer $t$.

######  Lemma 6.

Let $(\beta_{i_{1},i_{2}})_{1\leq i_{1}\leq m_{1},1\leq i_{2}\leq m_{2}}$ be
some elements of an ordered abelian group $G$, and let
$(\gamma_{1,j_{1}})_{j_{1}<\lambda_{1}}$,
$(\gamma_{2,j_{2}})_{j_{2}<\lambda_{2}}$ be two well-ordered, monotone
increasing sets of elements $\geq 0$ of $G$, without a last element. Let
$(t_{i_{1}})_{1\leq i_{1}\leq m_{1}}$, $(t_{i_{2}})_{1\leq i_{2}\leq m_{2}}$
be two sets of integers such that at least one of these sets contains only
nonzero integers and set $r=\operatorname{max}_{1\leq i_{1}\leq
m_{1}}|t_{i_{1}}|$. Assume that there exists $a\in G$ such that
$a>\gamma_{1,j_{1}}$ for all $j_{1}<\lambda_{1}$. Then there exist two
ordinals $\nu_{1}<\lambda_{1}$, $\nu_{2}<\lambda_{2}$ such that the elements
$P_{i_{1},i_{2},j_{1},j_{2},ra}=\beta_{i_{1},i_{2}}+t_{i_{1}}\gamma_{1,j_{1}}+t_{i_{2}}(\gamma_{2,j_{2}}+ra)$
for $1\leq i_{1}\leq m_{1},1\leq i_{2}\leq m_{2}$ are nonzero for any
$j_{1}>\nu_{1}$ and $j_{2}\geq\nu_{2}$.

###### Proof.

Consider the equations

$E_{i_{1},i_{2}}=\beta_{i_{1},i_{2}}+t_{i_{1}}x_{1}+t_{i_{2}}x_{2}=0$

for $1\leq i_{1}\leq m_{1},1\leq i_{2}\leq m_{2}$. For those equations $E$
with $t_{i_{2}}=0$ we see that $x_{1}=\gamma_{1,j_{1}}$ is not a solution for
all $j_{1}>\nu_{1}$, for a certain $\nu_{1}<\lambda_{1}$ which is obtained
easily using for example the proof of Lemma 5.

Now consider those above equations $E_{i_{1},i_{2}}$ when $t_{i_{2}}\not=0$.
It is enough to consider only those equations $E_{i_{1},i_{2}}$ such that

$P_{i_{1},i_{2},j_{1},j_{2},0}=\beta_{i_{1},i_{2}}+t_{i_{1}}\gamma_{1,j_{1}}+t_{i_{2}}\gamma_{2,j_{2}}=0$

for some $j_{1}>\nu_{1}$ and one $j_{2}$. If
$P_{i_{1},i_{2},j_{1}^{\prime},j_{2}^{\prime},ra}=\beta_{i_{1},i_{2}}+t_{i_{1}}\gamma_{1,j_{1}^{\prime}}+t_{i_{2}}(\gamma_{2,j_{2}^{\prime}}+ra)=0$
for some $j_{1}^{\prime}>\nu_{1}$ and $j_{2}^{\prime}>j_{2}$ then

$t_{i_{2}}(ra+\gamma_{2,j_{2}^{\prime}}-\gamma_{2,j_{2}})=t_{i_{1}}(\gamma_{1,j_{1}}-\gamma_{1,j_{1}^{\prime}})$

which is not possible because
$ra>t_{i_{1}}(\gamma_{1,j_{1}}-\gamma_{1,j_{1}^{\prime}})$,
$\gamma_{2,j_{2}^{\prime}}-\gamma_{2,j_{2}}>0$ and so

$|t_{i_{2}}(ra+\gamma_{2,j_{2}^{\prime}}-\gamma_{2,j_{2}})|>t_{i_{1}}(\gamma_{1,j_{1}}-\gamma_{1,j_{1}^{\prime}}).$

For each such equation $E_{i_{1},i_{2}}$ consider one such $j_{2}$ and let
$\nu_{2}$ be the biggest from these. Then $\nu_{1}$ and $\nu_{2}$ work.

######  Lemma 7.

Let $(\beta_{i_{1},i_{2}})_{1\leq i_{1}\leq m_{1},1\leq i_{2}\leq m_{2}}$ be
some elements of an ordered abelian group $G$, and let
$(\gamma_{1,j_{1}})_{j_{1}<\lambda_{1}}$,
$(\gamma_{2,j_{2}})_{j_{2}<\lambda_{2}}$ be two well-ordered, monotone
increasing sets of elements $\geq 0$ of $G$, without a last element. Let
$(t_{i_{1}})_{1\leq i_{1}\leq m_{1}}$, $(t_{i_{2}})_{1\leq i_{2}\leq m_{2}}$
be two sets of different integers and set $r=2\operatorname{max}_{1\leq
i_{1}\leq m_{1}}|t_{i_{1}}|$. Assume that there exists $a\in G$ such that
$a>\gamma_{1,j_{1}}$ for all $j_{1}<\lambda_{1}$. Then there exist two
ordinals $\nu_{1}<\lambda_{1}$, $\nu_{2}<\lambda_{2}$ such that the elements
$P_{i_{1},i_{2},j_{1},j_{2},ra}=\beta_{i_{1},i_{2}}+t_{i_{1}}\gamma_{1,j_{1}}+t_{i_{2}}(\gamma_{2,j_{2}}+ra)$
for $1\leq i_{1}\leq m_{1},1\leq i_{2}\leq m_{2}$ are different for any
$j_{1}>\nu_{1}$ and $j_{2}\geq\nu_{2}$.

###### Proof.

Consider the equations

$E_{i_{1},i_{2},i_{1}^{\prime},i^{\prime}_{2}}=\beta_{i_{1},i_{2}}-\beta_{i_{1}^{\prime},i_{2}^{\prime}}+(t_{i_{1}}-t_{i_{1}^{\prime}})x_{1}+(t_{i_{2}}-t_{i_{2}^{\prime}})x_{2}=0$

for all $(i_{1},i_{2})\not=(i_{1}^{\prime},i_{2}^{\prime})$. Note that above
we have either $t_{i_{1}}-t_{i_{1}^{\prime}}\not=0$, or
$t_{i_{2}}-t_{i_{2}^{\prime}}\not=0$ by hypothesis and in both cases their
absolute values are smaller than $r$. Let $\nu_{1}$, $\nu_{2}$ be given for
the equations $E_{i_{1},i_{2},i_{1}^{\prime},i^{\prime}_{2}}$ by the above
lemma. Then

$\beta_{i_{1},i_{2}}-\beta_{i_{1}^{\prime},i_{2}^{\prime}}+(t_{i_{1}}-t_{i_{1}^{\prime}})\gamma_{1,j_{1}}+(t_{i_{2}}-t_{i_{2}^{\prime}})(\gamma_{2,j_{2}}+ra)$

are nonzero for all $(i_{1},i_{2})\not=(i_{1}^{\prime},i_{2}^{\prime})$ and
any $j_{1}>\nu_{1}$, $j_{2}\geq\nu_{2}$, which is enough.

The following lemma extends Lemma 6.

######  Lemma 8.

Let $(\beta_{i_{1},\ldots,i_{n}})_{1\leq i_{1}\leq m_{1},\ldots,1\leq
i_{n}\leq m_{n}}$, $m_{e}\in{\bf N}$ be some elements of an ordered abelian
group $G$, and let $(\gamma_{e,j_{e}})_{j_{e}<\lambda_{e}}$, $1\leq e\leq n$,
be some well-ordered, monotone increasing sets of elements $\geq 0$ of $G$,
without last elements. Let $(t_{i_{e}})_{1\leq i_{e}\leq m_{e}}$, $1\leq e\leq
n$, be some sets of integers such that at least one of these sets contains
only nonzero integers and set $r_{e}=\operatorname{max}_{1\leq c\leq e,1\leq
i_{c}\leq m_{c}}|t_{i_{c}}|$, $1\leq e<n$. Assume that there exists $a_{e}\in
G$, $1\leq e<n$ such that $a_{e}>\gamma_{c,j_{c}}$ for all $1\leq c\leq e$,
$j_{c}<\lambda_{c}$. Then there exist some ordinals $\nu_{e}<\lambda_{e}$,
$1\leq e\leq n$ such that the elements

$P_{i_{1},\ldots,i_{n},j_{1},\ldots,j_{n},r_{1}a_{1},\ldots,r_{n-1}a_{n-1}}=\beta_{i_{1},\ldots,i_{n}}+t_{i_{1}}\gamma_{1,j_{1}}+\sum_{e=2}^{n}t_{i_{e}}(\gamma_{e,j_{e}}+r_{e-1}a_{e-1})$

are nonzero for $j_{e}>\nu_{e}$, $1\leq e\leq n$.

###### Proof.

Apply induction on $n$, the case $n=1,2$ being done in Lemma 5 and Lemma 6.
Assume that $n>2$ and consider the equations

$E_{i_{1},\ldots,i_{n}}=\beta_{i_{1},\ldots,i_{n}}+\sum_{e=1}^{n}t_{i_{e}}x_{e}=0$

for $1\leq i_{e}\leq m_{e}$, $1\leq e\leq n$. We follow the proof of Lemma 6.
For those equations $E_{i_{1},\ldots,i_{n}}$ with $t_{i_{n}}=0$ we see by
induction hypothesis that there exist $r_{e}$, $a_{e}$, $\nu_{e}<\lambda_{e}$,
$1\leq e<n$ such that

$P^{\prime}_{i_{1},\ldots,i_{n},j_{1},\ldots,j_{n-1},r_{1}a_{1},\ldots,r_{n-2}a_{n-2}}=\beta_{i_{1},\ldots,i_{n}}+t_{i_{1}}\gamma_{1,j_{1}}+\sum_{e=2}^{n-1}t_{i_{e}}(\gamma_{e,j_{e}}+r_{e-1}a_{e-1})$

are nonzero for $j_{e}>\nu_{e}$, $1\leq e\leq n-1$.

Consider the equations $E_{i_{1},\ldots,i_{n}}$ with $t_{i_{n}}\not=0$. As in
the proof of Lemma 6 it is enough to study only those equations
$E_{i_{1},\ldots,i_{n}}$ such that

$P_{i_{1},\ldots,i_{n},j_{1},\ldots,j_{n},r_{1}a_{1},\ldots,r_{n-2}a_{n-2},0}=\beta_{i_{1},\ldots,i_{n}}+t_{i_{1}}\gamma_{1,j_{1}}+\sum_{e=2}^{n-1}t_{i_{e}}(\gamma_{e,j_{e}}+r_{e-1}a_{e-1})+t_{i_{n}}\gamma_{n,j_{n}}=0$

for some $j_{1},\ldots,j_{n}$ with $j_{e}>\nu_{e}$, $e<n$. If

$P_{i_{1},\ldots,i_{n},j_{1}^{\prime},\ldots,j_{n}^{\prime},r_{1}a_{1},\ldots,r_{n-1}a_{n-1}}=\beta_{i_{1},\ldots,i_{n}}+t_{i_{1}}\gamma_{1,j_{1}}+\sum_{e=2}^{n}t_{i_{e}}(\gamma_{e,j_{e}^{\prime}}+r_{e-1}a_{e-1})=0$

for some $j_{1}^{\prime},\ldots,j_{n-1}^{\prime}$ with
$j_{e}^{\prime}>\nu_{e}$, $e<n$ and $j_{n}^{\prime}>j_{n}$ then

$t_{i_{n}}(r_{n-1}a_{n-1}+\gamma_{n,j_{n}^{\prime}}-\gamma_{n,j_{n}})=\sum_{e=1}^{n-1}t_{i_{e}}(\gamma_{e,j_{e}}-\gamma_{e,j_{e}^{\prime}})$

which is not possible because
$r_{n-1}a_{n-1}>t_{i_{c}}(\gamma_{c,j_{c}}-\gamma_{c,j_{c}^{\prime}})$ for all
$1\leq c<n$, $j_{c}<\lambda_{c}$,
$\gamma_{n,j_{n}^{\prime}}-\gamma_{n,j_{n}}>0$ and

$|t_{i_{n}}(r_{n-1}a_{n-1}+\gamma_{n,j_{n}^{\prime}}-\gamma_{n,j_{n}})|>\sum_{e=1}^{n-1}t_{i_{e}}(\gamma_{e,j_{e}}-\gamma_{e,j_{e}^{\prime}}).$

For each such equation $E_{i_{1},\ldots,i_{n}}$ consider one such $j_{n}$ and
let $\nu_{n}$ be the biggest from these. Then $\nu_{1},\ldots,\nu_{n}$ work.

The following lemma extends somehow Lemma 5.

######  Lemma 9.

Let $(\beta_{i_{1},\ldots,i_{n}})_{1\leq i_{1}\leq m_{1},\ldots,1\leq
i_{n}\leq m_{n}}$, $m_{e}\in{\bf N}$ be some elements of an ordered abelian
group $G$, and let $(\gamma_{e,j_{e}})_{j_{e}<\lambda_{e}}$, $1\leq e\leq n$,
be some well-ordered, monotone increasing sets of elements $\geq 0$ of $G$,
without last elements. Let $(t_{i_{e}})_{1\leq i_{e}\leq m_{e}}$, $1\leq e\leq
n$, be some sets of different integers and set
$r_{e}=2\operatorname{max}_{1\leq c\leq e,1\leq i_{c}\leq m_{c}}|t_{i_{c}}|$,
$1\leq e<n$. Assume that there exists $a_{e}\in G$, $1\leq e<n$ such that
$a_{e}>\gamma_{c,j_{c}}$ for all $1\leq c\leq e$, $j_{c}<\lambda_{c}$. Then
there exist some ordinals $\nu_{e}<\lambda_{e}$, $1\leq e\leq n$ such that the
elements

$P_{i_{1},\ldots,i_{n},j_{1},\ldots,j_{n},r_{1}a_{1},\ldots,r_{n-1}a_{n-1}}=\beta_{i_{1},\ldots,i_{n}}+t_{i_{1}}\gamma_{1,j_{1}}+\sum_{e=2}^{n}t_{i_{e}}(\gamma_{e,j_{e}}+r_{e-1}a_{e-1})$

are nonzero for $j_{e}>\nu_{e}$, $1\leq e\leq n$.

###### Proof.

Apply induction on $n$, the case $n=1,2$ being done in Lemma 5 and Lemma 7.
Assume $n>2$. As the proof of Lemma 7 consider the equations

$E_{i_{1},\ldots,i_{n},i_{1}^{\prime},\ldots,i^{\prime}_{n}}=\beta_{i_{1},\ldots,i_{n}}-\beta_{i_{1}^{\prime},\ldots,i_{n}^{\prime}}+\sum_{e=1}^{n}(t_{i_{e}}-t_{i_{e}^{\prime}})x_{i}=0$

for all $(i_{1},\ldots,i_{n})\not=(i_{1}^{\prime},\ldots,i_{n}^{\prime}).$
Note that above at least one from $t_{i_{e}}-t_{i_{e}^{\prime}}$ is nonzero by
hypothesis. Let $\nu_{1},\ldots,\nu_{n}$ be given for the equations
$E_{i_{1},\ldots,i_{n},i_{1}^{\prime},\ldots,i^{\prime}_{n}}$ by the above
lemma. Then

$\beta_{i_{1},\ldots,i_{n}}-\beta_{i_{1}^{\prime},\ldots,i_{n}^{\prime}}+(t_{i_{1}}-t_{i_{1}^{\prime}})\gamma_{1,j_{1}}+\sum_{e=2}^{n}(t_{i_{e}}-t_{i_{e}^{\prime}})(\gamma_{e,j_{e}}+r_{e-1}a_{e-1})$

are nonzero for all
$(i_{1},\ldots,i_{n})\not=(i_{1}^{\prime},\ldots,i_{n}^{\prime})$ and any
$j_{1}>\nu_{1}$,…, $j_{n}>\nu_{n}$, which is enough.

######  Lemma 10.

Let $V\subset V^{\prime}$ be an immediate extension of valuation rings,
$K\subset K^{\prime}$ its fraction field extension, $n\in{\bf N}$ and for any
$0\leq e\leq n$ the element $y_{e}\in V^{\prime}$ is a pseudo limit of a
pseudo convergent sequence $(v_{e,i_{e}})_{i_{e}<\lambda_{e}}$ over $V$, which
has no pseudo limit in $K$. Assume that there exist some polynomials $h_{e}\in
V[Y_{e}]$, $0\leq e\leq n$ such that $y_{e+1}=h_{e}(y_{e})$ for all $0\leq
e<n$ and $h_{e}$ has minimal degree among the polynomials $f\in V[Y_{e}]$ such
that $\mathrm{val}(f(v_{e,i_{e}}))<\mathrm{val}(f(v_{e,j_{e}}))$ for
$i_{e}<j_{e}<\lambda_{e}$ large (so $\operatorname{deg}h_{e}>1$). Set
$r_{s}=2\operatorname{max}_{0\leq e\leq s}\operatorname{deg}h_{e}-2$ and
suppose that for each $1\leq s<n$ there exists $b_{s}\in V$ such that
$\mathrm{val}(b_{s})>\mathrm{val}(v_{e,i_{e}})$ for all $0\leq e\leq s$ and
$i_{e}<\lambda_{e}$. Let $g\in V[Y_{0},\ldots,Y_{n}]$ be a polynomial with
$\operatorname{deg}_{Y_{e}}g<\operatorname{deg}h_{e}$ for $0\leq e\leq n$.
Then the following statements hold

  1. (1)

There exist some ordinals $\nu_{e}<\lambda_{e}$, $0\leq e\leq n$ such that the
elements ${\tilde{y}_{e}}=b_{e-1}^{r_{e-1}}y_{e}$, $2\leq e\leq n$,
${\tilde{y}_{1}}=y_{1}$ and the pseudo convergent sequences
$({\tilde{v}}_{e,i_{e}})_{i_{e}<\lambda_{e}}$ defined by
${\tilde{v}}_{e,i_{e}}=b_{e-1}^{r_{e-1}}v_{e,i_{e}}$,
${\tilde{v}}_{1,j_{1}}=v_{1,j_{1}}$ satisfy

$\mathrm{val}(g({\tilde{y}}_{0},\ldots,{\tilde{y}}_{n}))=\mathrm{val}(g({\tilde{v}}_{0,i_{0}},\ldots,{\tilde{v}}_{n,i_{n}}))$

for all $0\leq e\leq n$, $i_{e}>\nu_{e}$.

  2. (2)

Let
${\tilde{y}}_{e,i_{e}}=({\tilde{y}}_{e}-{\tilde{v}}_{e,i_{e}})/({\tilde{v}}_{e,i_{e}+1}-{\tilde{v}}_{e,i_{e}})$,
$0\leq e\leq n$. $i_{e}>\nu_{e}$. Then there exist $d\in V\setminus\\{0\\}$
and $u\in V[{\tilde{y}}_{0,i_{0}},\ldots,{\tilde{y}}_{n,i_{n}}]$ for some
$i_{e}>\nu_{e}$, $0\leq e\leq n$ with
$g({\tilde{y}}_{0},\ldots,{\tilde{y}}_{n})=du$ and $\mathrm{val}(u)=0$.

###### Proof.

For the first part apply induction on the total degree of $g$, the case
$\operatorname{deg}g=1$ follows for example using Lemma 2. Assume
$\operatorname{deg}g>1$. By Taylor’s formula we have

$g({\tilde{y}}_{0},\ldots,{\tilde{y}}_{n})=\sum_{l_{0},\ldots,l_{n}\geq
0}((D_{Y_{0}}^{(l_{0}))}\cdots
D_{Y_{n}}^{(l_{n})})(g))({\tilde{v}}_{0,i_{0}},\ldots.{\tilde{v}}_{n,i_{n}}))\cdot$
$({\tilde{y}}_{0}-{\tilde{v}}_{0,i_{0}})^{l_{0}}\cdots({\tilde{y}}_{n}-{\tilde{v}}_{n,i_{n}})^{l_{n}}$

with $((D_{Y_{0}}^{(l_{0})}\cdots D_{Y_{n}}^{(l_{n})}(g))\in
V[Y_{0},\ldots,Y_{n}]$ be the partial derivations of $g$ with respect to
$Y_{0},\ldots,Y_{n}$. By induction hypothesis let the ordinals
$\nu_{e}^{\prime}<\lambda_{e}$, $0\leq e\leq n$ be given for
$((D_{Y_{0}}^{(l_{0})}\cdots D_{Y_{n}}^{(l_{n})}(g))$, $l_{0}+\ldots+l_{n}>0$,
such that

$\mathrm{val}((D_{Y_{0}}^{(l_{0})}\cdots
D_{Y_{n}}^{(l_{n})})(g)({\tilde{v}}_{0,i_{0}},\ldots,{\tilde{v}}_{n,i_{n}}))=\mathrm{val}((D_{Y_{0}}^{(l_{0})}\cdots
D_{Y_{n}}^{(l_{n})})(g)({\tilde{y}}_{0},\ldots,{\tilde{y}}_{n}))$

are constant for $l_{0}+\ldots+l_{n}>0$ when $i_{e}>\nu_{e}^{\prime}$, $0\leq
e\leq n$ (note that $r_{s},b_{s}$ and so
${\tilde{y}}_{e},{\tilde{v}}_{e,j_{e}}$ do not depend on $g$).

Apply Lemma 9 for
$\beta_{l_{0},\ldots,l_{n}}=\mathrm{val}((D_{Y_{0}}^{(l_{0})}\cdots
D_{Y_{n}}^{(l_{n})})(g)({\tilde{y}}_{0},\ldots,{\tilde{y}}_{n}))$,
$t_{l_{e}}=l_{e}$ and
$\gamma_{e,j_{e}}=\mathrm{val}({\tilde{y}}_{e}-{\tilde{v}}_{e,j_{e}})=\mathrm{val}({\tilde{v}}_{e,j_{e}+1}-{\tilde{v}}_{e,j_{e}})$,
$0\leq e\leq n$. Then there exist some ordinals
$\nu_{e}^{\prime}<\nu_{e}<\lambda_{e}$, $0\leq e\leq n$ such that
$\mathrm{val}(g({\tilde{y}}_{0},\ldots,{\tilde{y}}_{n}))$ and the valuations
of

$((D_{Y_{0}}^{(l_{0})}\cdots
D_{Y_{n}}^{(l_{n})})(g))({\tilde{y}}_{0},\ldots,{\tilde{y}}_{n})\cdot({\tilde{y}}_{0}-{\tilde{v}}_{0,i_{0}})^{l_{0}}\cdots({\tilde{y}}_{n}-{\tilde{v}}_{n,i_{n}})^{l_{n}},$

for $l_{0}+\ldots+l_{n}>0$, with $((D_{Y_{0}}^{(l_{0})}\cdots
D_{Y_{n}}^{(l_{n})})(g))({\tilde{y}}_{0},\ldots,{\tilde{y}}_{n})\not=0$ are
all different for $i_{e}>\nu_{e}$, $0\leq e\leq n$. Therefore from the
Taylor’s formula we have either
$\mathrm{val}(g({\tilde{y}}_{0},\ldots,{\tilde{y}}_{n}))=\mathrm{val}(g({\tilde{v}}_{0,i_{0}},\ldots,{\tilde{v}}_{n,i_{n}}))$
for $i_{e}>\nu_{e}$, $0\leq e\leq n$, which we need, or
$\mathrm{val}(g({\tilde{v}}_{0,i_{0}},\ldots,{\tilde{v}}_{n,i_{n}}))=$

$\mathrm{val}((D_{Y_{0}}^{(l_{0})}\cdots
D_{Y_{n}}^{(l_{n})})(g))({\tilde{v}}_{0,i_{0}},\ldots,{\tilde{v}}_{n,i_{n}})\cdot({\tilde{y}}_{0}-{\tilde{v}}_{0,i_{0}})^{l_{0}}\cdots({\tilde{y}}_{n}-{\tilde{v}}_{n,i_{n}})^{l_{n}}),$

for some $l_{0},\ldots,l_{n}$ with $l_{0}+\ldots+l_{n}>0$ depending on
$i_{0},\ldots,i_{n}$. As the set of these $l_{0},\ldots,l_{n}$ is finite we
may assume that $l_{e}>0$, for $i_{e}>\nu_{e}$, $0\leq e\leq n$ with some
gaps, namely for each $\nu_{e}<\tau_{e}<\lambda_{e}$ there exists
$i_{e}>\tau_{e}$ such that the above equality holds independently on
$i_{j}>\nu_{j}$, $0\leq j\leq n$, $j\not=e$.

The second possibility is not possible. Fix $i_{j}>\nu_{j}$, $0\leq j\leq n$,
$j\not=e$. Then
$\mathrm{val}(g({\tilde{v}}_{0,i_{0}},\ldots,{\tilde{v}}_{n,i_{n}}))=$

$\mathrm{val}((D_{Y_{0}}^{(l_{0})}\cdots
D_{Y_{n}}^{(l_{n})})(g))({\tilde{v}}_{0,i_{0}},\ldots,{\tilde{v}}_{n,i_{n}})\cdot({\tilde{y}}_{0}-{\tilde{v}}_{0,i_{0}})^{l_{0}}\cdots({\tilde{y}}_{n}-{\tilde{v}}_{n,i_{n}})^{l_{n}})$

is increasing for $i_{e}>\nu_{e}$ with some gaps as above. So for each
$\nu_{e}<\tau_{e}<\lambda_{e}$ the polynomial

$F(Y_{e})=g({\tilde{v}}_{0,i_{0}},\ldots,{\tilde{v}}_{e-1,i_{e-1}},Y_{e},{\tilde{v}}_{e+1,i_{e+1}},\ldots,{\tilde{v}}_{n,i_{n}})$

of degree $<\operatorname{deg}h_{e}$ satisfies
$\mathrm{val}(F({\tilde{v}}_{e,i_{e}}))<\mathrm{val}(F({\tilde{v}}_{e,i_{e}^{\prime}}))$
for some $\tau_{e}<i_{e}<i_{e}^{\prime}<\lambda_{e}$. But combining [7, Lemmas
1, 5] we see that this gives in fact
$\mathrm{val}(F({\tilde{v}}_{e,i_{e}}))<\mathrm{val}(F({\tilde{v}}_{e,i_{e}^{\prime}}))$
for $i_{e}<i_{e}^{\prime}<\lambda_{e}$ large enough, which is false.

Thus (1) holds and so

$\mathrm{val}(g({\tilde{y}}_{0},\ldots,{\tilde{y}}_{n}))=\mathrm{val}(g({\tilde{v}}_{0,i_{0}},\ldots,{\tilde{v}}_{n,i_{n}}))<$
$\mathrm{val}((D_{Y_{0}}^{(l_{0})}\cdots
D_{Y_{n}}^{(l_{n})})(g))({\tilde{v}}_{0,i_{0}},\ldots,{\tilde{v}}_{n,i_{n}})\cdot({\tilde{y}}_{0}-{\tilde{v}}_{0,i_{0}})^{l_{0}})\cdots({\tilde{y}}_{n}-{\tilde{v}}_{n,i_{n}})^{l_{n}})$

for all $l_{0},\ldots,l_{n}$ with $l_{0}+\ldots+l_{n}>0$, when
$i_{0},\ldots,i_{n}$ are enough large.

It follows that

$g({\tilde{y}}_{0},\ldots,{\tilde{y}}_{n})=\sum_{l_{0},\ldots,l_{n}\geq
0}((D_{Y_{0}}^{(l_{0})}\cdots
D_{Y_{n}}^{(l_{n})})(g))({\tilde{v}}_{0,i_{0}},\ldots.{\tilde{v}}_{n,i_{n}}))\cdot$
$({\tilde{v}}_{0,i_{0}+1}-{\tilde{v}}_{0,i_{0}})^{l_{0}}\cdots({\tilde{v}}_{n,i_{n}+1}-{\tilde{v}}_{n,i_{n}})^{l_{n}}{\tilde{y}}_{0,i_{0}}^{l_{0}}\cdots{\tilde{y}}_{n,i_{n}}^{l_{n}}$
$\in
g({\tilde{v}}_{0,i_{0}},\ldots,{\tilde{v}}_{n,j_{n}})(1+{\mathfrak{m}}^{\prime}\cap
V[{\tilde{y}}_{0,i_{0}},\ldots,{\tilde{y}}_{n,j_{n}}]),$

which is enough for the second part, ${\mathfrak{m}}^{\prime}$ being the
maximal ideal of $V^{\prime}$.

######  Theorem 11.

Let $V^{\prime}$ be an immediate extension of a valuation ring $V$, $K\subset
K^{\prime}$ their fraction field extension, $v=(v_{j})_{j<\lambda}$ an
algebraic pseudo-convergent sequence in $V$ with a pseudo limit $x$ in
$V^{\prime}$, but having no pseudo limit in $K$. Assume that $x$ is
transcendental over $K$ and $K^{\prime}=K(x)$. Then $V^{\prime}$ is either a
filtered union of its complete intersection $V$-subalgebras, or there exists
an element $y\in K^{\prime}\setminus K$ which is a pseudo limit of a
transcendental pseudo convergent sequence in $V$, which has no pseudo limit in
$V$.

###### Proof.

Let $h_{0}$ be a primitive polynomial in $V[X]$ of minimal degree among the
primitive polynomials $f$ such that
$\mathrm{val}(f(v_{i}))<\mathrm{val}(f(v_{j}))$, for large enough
$i<j<\lambda$. We must have $\operatorname{deg}h_{0}>1$ because otherwise $v$
has a pseudo limit in $V$, which is false. Set $x_{0}=x$,
$\lambda_{0}=\lambda$, $a_{0,j_{0}}=v_{j_{0}}$. By induction on $n$ we
construct a sequence of elements $x_{n}\in V^{\prime}$, $n\geq 1$, which are
pseudo limits of some pseudo convergent sequences
$a_{n}=(a_{n,j_{n}})_{j_{n}<\lambda_{n}}$, $n\geq 1$ over $V$, with no pseudo
limits in $V$ (in case they are not fundamental) and some polynomials
$h_{n}\in V[X]$ with $\operatorname{deg}h_{n}>1$.

If $a_{n}$ is transcendental or it is fundamental then the construction stops
and we may take $y=x_{n}$. If $a_{n}$ is fundamental then it follows also that
it is transcendental because $x_{n}$ is transcendental over $K$. Indeed, if
$f\in V[X]\setminus\\{0\\}$ then $f(x_{n})\not=0$ and we see that
$\mathrm{val}(a_{n,j_{n}+1}-a_{n,j_{n}})>\mathrm{val}(f(x_{n}))$ for $j_{n}$
enough large, $a_{n}$ being fundamental. It follows that
$\mathrm{val}(f(x_{n})-f(a_{n,j_{n}}))\geq\mathrm{val}(x_{n}-a_{n,j_{n}})>\mathrm{val}(f(x_{n}))$
and so $\mathrm{val}(f(x_{n}))=\mathrm{val}(f(a_{n,j_{n}}))$ for $j_{n}$
enough large.

Assume that $a_{n}$ is algebraic. Then $h_{n}$ is chosen to be a primitive
polynomial of minimal degree among all polynomials $f\in V[X]$ such that
$\mathrm{val}(f(a_{n,i_{n}}))<\mathrm{val}(f(a_{n,j_{n}}))$ for all large
enough $i_{n}<j_{n}<\lambda_{n}$. Set $x_{n+1}=h_{n}(x_{n})$ and using [7,
Theorem 1] choose $a_{n+1}=(a_{n+1,j_{n+1}})_{j_{n+1}<\lambda_{n+1}}$ over $V$
which has no pseudo limit in $V$ but for which $x_{n+1}$ is a pseudo limit in
$V^{\prime}$.

As we saw we may assume that all $a_{n}$, $n\in{\bf N}$ are algebraic and not
fundamental. For all $n$ and each $j_{n}<\lambda_{n}$, we set

$x_{n,j_{n}}=(x_{n}-a_{n,j_{n}})/(a_{n,j_{n}+1}-a_{n,j_{n}}),$

so that, $x_{n,j_{n}}$ is an unit in $V^{\prime}$. By Lemma 10 we consider
$r_{n}\in{\bf N}$, $b_{n}\in V$ defined for $a_{n,j_{n}}$ (here it is useful
that $a_{n}$ is not fundamental). Changing $x_{n}$ by ${\tilde{x}}_{n}$,
$x_{n,j_{n}}$ by ${\tilde{x}}_{n,j_{n}}$ (actually
$x_{n,j_{n}}={\tilde{x}}_{n,j_{n}}$ for all $j_{n}>1$) and $a_{n,j_{n}}$ by
${\tilde{a}}_{n,j_{n}}$ defined as in Lemma 10 we get

$F(x_{0},\ldots,x_{n}))\in
F(a_{0,j_{0}},\ldots,a_{n,j_{n}})(1+{\mathfrak{m}}^{\prime}\cap
V[x_{0},\ldots,x_{n}])$

for every $F\in V[X_{0},\ldots,X_{n}]$ with
$\operatorname{deg}_{Y_{e}}F<\operatorname{deg}h_{e}$, $1\leq e\leq n$ and
$j_{0},\ldots,j_{n}$ enough large, ${\mathfrak{m}}^{\prime}$ being the maximal
ideal of $V^{\prime}$.

Let $A_{j_{0},\ldots,j_{n}}$ be the localization of
$V[x_{0,j_{0}},\ldots,x_{n,j_{n}}]$ in ${\mathfrak{m}}^{\prime}\cap
V[x_{0,j_{0}},\ldots,x_{n,j_{n}}]$. Set

$q_{n,j_{n}}=a_{n,j_{n}}+(a_{n,j_{n}+1}-a_{n,j_{n}})X_{n,j_{n}}\in
V[X_{n,j_{n}}],\ \ \ h_{n,j_{n}}=h_{n}(q_{n,j_{n}}),$
$g_{j_{n},j_{n+1}}=h_{n,j_{n}}-q_{n+1,j_{n+1}}\in
V[X_{n,j_{n}},X_{n+1,j_{n+1}}]$

for $n\geq 0$,
$C_{j_{0},\ldots,j_{n}}=V[X_{0,j_{0}},\ldots,X_{n,j_{n}}]/(g_{j_{0},j_{1}},\ldots,g_{j_{n-1},j_{n}})$
and let
$\rho_{j_{0},\ldots,j_{n}}:C_{j_{0},\ldots,j_{n}}\rightarrow
A_{j_{0},\ldots,j_{n}}$ be the map given by $X_{i,j_{i}}\rightarrow
x_{i,j_{i}}$. We have

$q_{n,j_{n}}(x_{n,j_{n}})=x_{n}=h_{n-1}(x_{n-1})=h_{n-1,j_{n-1}}(x_{n-1.j_{n-1}}),$

that is $g_{j_{n-1},j_{n}}(x_{n-1,j_{n-1}},x_{n.j_{n}})=0$.

Note that

$(a_{n,j_{n}+1}-a_{n,j_{n}})x_{n,j_{n}}=(a_{n,j_{n}^{\prime}}-a_{n,j_{n}})+(a_{n,j_{n}^{\prime}+1}-a_{n,j_{n}^{\prime}})x_{n,j_{n}^{\prime}}$

and as

$\mathrm{val}(a_{n,j_{n}^{\prime}}-a_{n,j_{n}})=\mathrm{val}(a_{n,j_{n}+1}-a_{n,j_{n}})<\mathrm{val}(a_{n,j_{n}^{\prime}+1}-a_{n,j_{n}^{\prime}})$

for $j_{n}<j_{n^{\prime}}$ we get
$x_{n,j_{n}}=\alpha_{j_{n},j_{n}^{\prime}}+\beta_{j_{n},j_{n}^{\prime}}x_{n,j_{n}^{\prime}}$
for some unit $\alpha_{j_{n},j_{n}^{\prime}}\in V$ and non unit
$\beta_{j_{n},j_{n}^{\prime}}\in V$. Thus $A_{j_{0},\ldots,j_{n}}\subset
A_{j_{0}^{\prime},\ldots,j_{n}^{\prime}}$ if $j_{i}\leq j_{i}^{\prime}$,
$0\leq i\leq n$ and so the union $A$ of $A_{j_{0},\ldots,j_{n}}$ is filtered.
Similarly we have some canonical maps $C_{j_{0},\ldots,j_{n}}\rightarrow
C_{j_{0}^{\prime},\ldots,j_{n}^{\prime}}$, $j_{i}\leq j_{i}^{\prime}$, $0\leq
i\leq n$ given by
$X_{i,j_{i}}\rightarrow(\alpha_{j_{i},j_{i}^{\prime}}+\beta_{j_{i},j_{i}^{\prime}}X_{i,j_{i}^{\prime}})$.
These give together with the canonical maps $C_{j_{0},\ldots,j_{n}}\rightarrow
C_{j_{0},\ldots,j_{n+1}}$ a filtered direct system and there exists a map
$\rho$ from its limit $C$ to $A$ extending $\rho_{j_{0},\ldots,j_{n}}$.
Clearly, $C$ is a filtered direct limit of some complete intersection
$V$-algebras.

Let $f\in V[X_{0}]$. We claim that we may get

$cf=F(X_{0},h_{0}(X_{0}),\ldots,h_{n-1}(\ldots(h_{0}(X_{0}))\ldots))$

for some $n\in{\bf N}$, $c\in V\setminus\\{0\\}$ and $F\in
V[X_{0},\ldots,X_{n}]$ with
$\operatorname{deg}_{X_{i}}F<\operatorname{deg}_{X_{i}}h_{i}$ for any $0\leq
i<n$ if $a_{0},\ldots,a_{n}$ are algebraic pseudo convergent sequences.
Indeed, by division with $h_{0}$ we get $c^{\prime}f=g_{1}h_{0}+g_{0}$ for
some $c^{\prime}\in V\setminus\\{0\\}$ and some polynomials $g_{1},g_{0}\in
V[X_{0}]$ such that $\operatorname{deg}g_{1}<\operatorname{deg}f$ and
$\operatorname{deg}g_{0}<\operatorname{deg}h_{0}$. Repeating this procedure
dividing $g_{1}$ by $h_{0}$ we get finally $c_{0}f=f_{1}(X_{0},h_{0}(X_{0}))$
for some $c_{0}\in V\setminus\\{0\\}$ and some polynomial $f_{1}\in
V[X_{0},X_{1}]$ with
$\operatorname{deg}_{X_{0}}f_{1}<\operatorname{deg}h_{0}$. By the same method
dividing $f_{1}$ (considered in $V[X_{0}][X_{1}]$) by $h_{1}$ we get
$cf_{1}=g_{1}^{\prime}h_{1}+g^{\prime}_{0}$ for some $c\in V\setminus\\{0\\}$
and some polynomials $g^{\prime}_{1},g^{\prime}_{0}\in V[X_{0},X_{1}]$ with
$\operatorname{deg}_{X_{1}}g^{\prime}_{1}<\operatorname{deg}_{X_{1}}f_{1}$,
$\operatorname{deg}_{X_{1}}g^{\prime}_{0}<\operatorname{deg}_{X_{1}}h_{1}$.
Repeating this procedure dividing $g^{\prime}_{1}$ by $h_{1}$ we get finally
$c_{1}f_{1}=f_{2}(X_{0},X_{1},h_{1}(X_{1}))$ for some $c_{1}\in
V\setminus\\{0\\}$ and $f_{2}\in V[X_{0},X_{1},X_{2}]$ with
$\operatorname{deg}_{X_{1}}f_{2}<\operatorname{deg}_{X_{1}}h_{1}$ and such
that $c_{0}c_{1}f=f_{2}(X_{0},h_{0}(X_{0}),h_{1}(h_{0}(X_{0}))$. After several
steps, the claim is stated.

Actually, the above procedure stops for a $n$ big enough (depending on
$\operatorname{deg}f$) to have
$\operatorname{deg}_{X_{i}}F<\operatorname{deg}_{X_{i}}h_{i}$ for all $0\leq
i\leq n$. Thus $F(x_{0},\ldots,x_{n})=du$ for some $d\in V\setminus\\{0\\}$
and $u\in V[x_{0,j_{0}},\ldots,x_{n,j_{n}}]$ with $\mathrm{val}(u)=0$ using
Lemma 10. In fact we have $f(x_{0})=d^{\prime}u$ for $d^{\prime}=d/c\in
V\setminus\\{0\\}$. It follows that $A=V^{\prime}$. Indeed, let $f_{1}/f_{2}$
be a rational function from $V^{\prime}$. As above for $n=0$ we may assume
$f_{i}(x_{0})=d_{i}u_{i}$, $i=1,2$ for some $d_{i}\in V\setminus\\{0\\}$ and
some units $u_{i}\in A$. Then $\mathrm{val}(d_{1})\geq\mathrm{val}(d_{2})$
because $f_{1}/f_{2}\in V^{\prime}$ and
$f_{1}/f_{2}=(d_{1}/d_{2})u_{1}u_{2}^{-1}\in A$.

Remains to show that $A$ is a filtered union of complete intersection
$V$-subalgebras. It will be enough to show that $\rho$ is injective. Consider
the ideal

$I=(X_{1}-h_{0}(X_{0}),\ldots,X_{n}-h_{n-1}(X_{n-1}),\ldots)$

of $V[X_{0},\ldots,X_{n},\ldots]$. As above for all $f\in
V[X_{0},\ldots,X_{n}]$ there exist $c\in V\setminus\\{0\\}$ and $F\in
V[X_{0},\ldots,X_{r}]$ for some $r\geq n$ such that
$\operatorname{deg}_{X_{i}}F<\operatorname{deg}_{X_{i}}h_{i}$ for some $r\geq
n$ and $cf\equiv F$ modulo $I$. On the other hand as in the proof of Lemma 10
we see using Taylor’s formula that

$F(x_{0},\ldots,x_{r})=F(x_{0},\ldots,x_{r-1},a_{r,j_{r}})+\sum_{l_{r}>0}(D^{(l_{r})}F)(x_{0},\ldots,x_{r-1},a_{r,j_{r}})(x_{r}-a_{r,j_{r}})^{l_{r}}$

with

$\mathrm{val}(F(x_{0},\ldots,x_{r-1},a_{r,j_{r}}))<\mathrm{val}((D^{(l_{r})}F)(x_{0},\ldots,x_{r-1},a_{r,j_{r}})(x_{r}-a_{r,j_{r}})^{l_{r}})$

for $l_{r}>0$ and

$\mathrm{val}(F(x_{0},\ldots,x_{r}))=\mathrm{val}((F(x_{0},\ldots,x_{r-1},a_{r,j_{r}}))=\mathrm{val}((F(a_{0,j_{0}},\ldots,a_{r-1,j_{r-1}},a_{r,j_{r}}))$

for some $j_{0},\ldots,j_{r}$. Using again Taylor’s formula for $F$ and
$D^{(l_{r})}F$ we get

$F(x_{0},\ldots,x_{r})=F(a_{0,j_{0}},\ldots,a_{r,j_{r}})+$
$\sum_{l_{0},\ldots,l_{r}\geq 0,l_{0}+\ldots
l_{r}>0}(D_{X_{0}}^{(l_{0}))}\cdots
D_{X_{r}}^{(l_{r})})(F)(a_{0,j_{0}},\ldots,a_{r,j_{r}})\cdot$
$(x_{0}-a_{0,j_{0}})^{l_{0}}\ldots(x_{r-1}-a_{r-1,j_{r-1}})^{l_{r-1}}(x_{r}-a_{r,j_{r}})^{l_{r}}$

with

$\mathrm{val}(F(a_{0,j_{0}},\ldots,a_{r,j_{r}}))<\mathrm{val}((D_{X_{0}}^{(l_{0}))}\cdots
D_{X_{r}}^{(l_{r})})(F)(a_{0,j_{0}},\ldots,a_{r,j_{r}})\cdot$
$(x_{0}-a_{0,j_{0}})^{l_{0}}\ldots(x_{r-1}-a_{r-1,j_{r-1}})^{l_{r-1}}(x_{r}-a_{r,j_{r}})^{l_{r}})$

for $l_{0},\ldots,l_{r}\geq 0,l_{0}+\ldots l_{r}>0$. It follows that
$F(q_{0,j_{0}},\ldots,q_{r,j_{r}})=$

$F(a_{0,j_{0}},\ldots,a_{r,j_{r}})+\sum_{l_{0},\ldots,l_{r}\geq 0,l_{0}+\ldots
l_{r}>0}((D_{X_{0}}^{(l_{0}))}\cdots
D_{X_{r}}^{(l_{r})})(F))(a_{0,j_{0}},\ldots,a_{r,j_{r}})\cdot$
$(a_{0,j_{0}+1}-a_{0,j_{0}})^{l_{0}}\cdots(a_{r,j_{r}+1}-a_{r,j_{r}})^{l_{r}}X_{0,j_{0}}^{l_{0}}\cdots
X_{r,j_{r}}^{l_{r}}=dU$

for $d=F(a_{0,j_{0}},\ldots a_{r,j_{r}})$ and

$U=1+(1/d)\sum_{l_{0},\ldots,l_{r}\geq 0,l_{0}+\ldots
l_{r}>0}((D_{X_{0}}^{(l_{0}))}\cdots
D_{X_{r}}^{(l_{r})})(F))(a_{0,j_{0}},\ldots,a_{r,j_{r}})\cdot$
$(a_{0,j_{0}+1}-a_{0,j_{0}})^{l_{0}}\cdots(a_{r,j_{r}+1}-a_{r,j_{r}})^{l_{y}}X_{0,j_{0}}^{l_{0}}\cdots
X_{r,j_{r}}^{l_{r}}.$

Thus $F(q_{0,j_{0}},\ldots,q_{r,j_{r}})\not\in\operatorname{Ker}\rho$.

An element of $C$ is induced by a polynomial $P\in
V[X_{0,j_{0}},\ldots,X_{n,j_{n}}]$ and there exist $c^{\prime}\in
V\setminus\\{0\\}$ and $Q\in V[X_{0},\ldots,X_{n}]$ such that
$c^{\prime}P=Q(q_{0,j_{0}},\ldots,q_{n,j_{n}})$. As above there exist
$c^{\prime\prime}\in V\setminus\\{0\\}$ and $F\in V[X_{0},\ldots,X_{r}]$ for
some $r\geq n$ such that
$\operatorname{deg}_{X_{i}}F<\operatorname{deg}_{X_{i}}h_{i}$ for $i\leq r$
and $c^{\prime\prime}Q\equiv F$ modulo $I$. Then
$c^{\prime\prime}Q(q_{0,j_{0}},\ldots,q_{n,j_{n}})\equiv
F(q_{0,j_{0}},\ldots,q_{r,j_{r}})$ modulo
$(g_{j_{0},j_{1}},\ldots,g_{j_{r-1},j_{r}})$. It follows that
$c^{\prime}c^{\prime\prime}P$ does not induce an element from
$\operatorname{Ker}\rho$ if it is not in
$(g_{j_{0},j_{1}},\ldots,g_{j_{r-1},j_{r}})$. Hence $\rho$ is injective.

##  2\. Algebraic immediate extensions of valuation rings

The following lemma is given by Ostrowski ([12, page 371, IV and III], see
also [17, (II,4), Lemma 9]) and we need it next.

######  Lemma 12.

If $\\{v_{j}\\}$ is a pseudo convergent sequence in $K$, and $f\in K[X]$ is a
polynomial, then $\\{f(v_{j})\\}$ is ultimately pseudo convergent.

We also need the following lemma which follows from [10, Lemma 5.2].

######  Lemma 13.

Assume that $(v_{j})_{j}$ is a pseudo-convergent sequence in $V$ with a pseudo
limit $x$ in an extension $V^{\prime}$ of $V$ but with no pseudo limit in $V$.
Let $f\in V[X]$ be a polynomial. Then $(f(v_{j}))_{j}$ is ultimately pseudo-
convergent and $f(x)$ is a pseudo limit of it.

######  Proposition 14.

Let $V^{\prime}$ be an immediate extension of a valuation ring $V$ and
$K\subset K^{\prime}$ their fraction field extension and $K^{\prime}=K(x)$ for
some $x\in K^{\prime}$ algebraic over $K$. Then $V^{\prime}$ is a filtered
union of its complete intersection $V$-subalgebras.

###### Proof.

By [7, Theorem 1] there exists an algebraic pseudo-convergent sequence
$(v_{j})_{j<\lambda}$ in $V$ having $x$ as a pseudo limit but no pseudo limits
in $K$.

We claim that there exists an extension $K\subset K^{\prime\prime}$ contained
strictly in $K(x)$ such that $g=$Irr$(x,K^{\prime\prime})$ has minimal degree
among all polynomials $f\in K^{\prime\prime}[Y]$ such that
$\mathrm{val}(f(v_{j}))<\mathrm{val}(f(v_{j+1}))$ for large $j<\lambda$. Let
$h\in V[Y]$ be a polynomial with minimal degree among the polynomials $f\in
V[Y]$ such that $\mathrm{val}(f(v_{i}))<\mathrm{val}(f(v_{j}))$ for large
$i<j<\lambda$. Note that $\operatorname{deg}h>1$ because $(v_{j})_{j<\lambda}$
has no pseudo limits in $K$.

We see that $(v_{j})_{j<\lambda}$ has no pseudo limit in $K(h(x))$. Indeed,
assume that $q(h(x))$, $q\in V[Y]$ is such pseudo limit in $K(h(x))$. We have
$\mathrm{val}(v_{j})=\mathrm{val}(v_{j+1})=\ldots$ because
$\operatorname{deg}h>1$,
$\mathrm{val}(h(v_{j})<\mathrm{val}(h(v_{j+1}))<\ldots$ and
$\mathrm{val}(h(x)>\mathrm{val}(h(v_{j}))$ for $j$ large because otherwise we
have $\mathrm{val}(h(x))<\mathrm{val}(h(v_{r}))$ for some $r$ large and we get

$\mathrm{val}(h(x)-h(v_{r}))=\mathrm{val}(h(x))<\mathrm{val}(h(v_{r}))=\mathrm{val}(h(v_{r+1})-h(v_{r}))$

which shows that $h(x)$ is not a pseudo limit of $h(v_{j})_{j}$ contradicting
Lemma 13. Then

$\mathrm{val}(q(h(x))-q(0))\geq\mathrm{val}(h(x))>\mathrm{val}(h(v_{j}))=$
$\mathrm{val}(h(v_{j+1})-h(v_{j}))\geq\mathrm{val}(v_{j+1}-v_{j}).$

If $q(0)\not=0$ then $q(0)\in K$ is a pseudo limit of $(v_{j})_{j}$, which is
false. If $q(0)=0$ then choose $b\in K^{*}$ such that
$\mathrm{val}(b)>\mathrm{val}(q(h(x)))$ and we have
$\mathrm{val}(b-q(h(x)))=\mathrm{val}(q(h(x)))>\mathrm{val}(v_{j+1}-v_{j})$
for all $j$ and so $b$ is a pseudo limit of $v$ in $K$, which is again
impossible.

If $h$ has minimal degree among the polynomials $f\in K(h(x))[Y]$ such that
$\mathrm{val}(f(v_{i}))<\mathrm{val}(f(v_{j}))$ for large $i<j<\lambda$ then
$\operatorname{deg}h\leq\operatorname{deg}g_{1}$, where $g_{1}$ is
Irr$(x,K(h(x)))$ and must be also such $f$ by Lemma 3 (1). But $x$ is a root
of $g_{1}^{\prime}=h(Y)-h(x)\in K(h(x))[Y]$ and so $g_{1}$ divides
$g_{1}^{\prime}$. Thus $g_{1}=g_{1}^{\prime}$ because
$\operatorname{deg}g^{\prime}_{1}=\operatorname{deg}h\leq\operatorname{deg}g_{1}$
and $K^{\prime\prime}=K(h(x))$, $g_{1}$ work, that is $g_{1}$ has minimal
degree among all polynomials $f\in K^{\prime\prime}[Y]$ such that
$\mathrm{val}(f(v_{j}))<\mathrm{val}(f(v_{j+1}))$ for large $j<\lambda$.
Otherwise, we may continue with $K(h(x))$ instead $K$, taking $h_{1}\in
K(h(x))[Y]$ to be a polynomial of minimal degree among the polynomials $f\in
K(h(x))[Y]$ such that $\mathrm{val}(f(v_{i}))<\mathrm{val}(f(v_{j}))$ for
large $i<j<\lambda$ and consider the extension $K\subset K(h_{1}(x),h(x))$. If
$h_{1}$ has minimal degree among the polynomials $f\in K(h_{1}(x),h(x))[Y]$
such that $\mathrm{val}(f(v_{i}))<\mathrm{val}(f(v_{j}))$ for large
$i<j<\lambda$ then $K^{\prime\prime}=K(h_{1}(x),h(x))$ works. Note that
$\operatorname{deg}h_{1}<\operatorname{deg}h$. Also
$\operatorname{deg}h_{1}>1$ because if $\operatorname{deg}h_{1}=1$ then
$(v_{j})_{j<\lambda}$ has a pseudo limit in $K(h(x)$ and we saw above that
this not the case.

We may continue to find an extension $K\subset K^{\prime\prime}$, contained
strictly in $K(x)$ and
$1<\operatorname{deg}h_{s}<\dots<\operatorname{deg}h_{1}<\operatorname{deg}h$
such that $g_{s}=$Irr$(x,K^{\prime\prime})$ has minimal degree, that is
$\operatorname{deg}h_{s}$, among all polynomials $f\in K^{\prime\prime}[Y]$
such that $\mathrm{val}(f(v_{j}))<\mathrm{val}(f(v_{j+1}))$ for large
$j<\lambda$, which shows our claim. Then $V^{\prime}$ is a filtered union of
its complete intersection $V^{\prime\prime}=V^{\prime}\cap
K^{\prime\prime}$-subalgebras by Lemma 3 (2). Use induction on
$[K^{\prime}:K]$. By induction hypothesis we see that $V^{\prime\prime}$ is a
filtered union of its complete intersection $V$-subalgebras. Then $V^{\prime}$
is also a filtered union of its complete intersection $V$-subalgebras.

######  Proposition 15.

Let $V^{\prime}$ be an immediate extension of a valuation ring $V$, $K\subset
K^{\prime}$ their fraction field extension, such that $K^{\prime}=K(x)$ for
some $x\in K^{\prime}$. Then $V^{\prime}$ is a filtered union of its complete
intersection $V$-subalgebras.

###### Proof.

If $x$ is algebraic over $K$ apply Proposition 14. Assume that $x$
transcendental over $K$. By Theorem 11 remains to see the case when there
exists $y\in K^{\prime}$ such that $y$ is a pseudo limit of a transcendental
pseudo convergent sequence over $V$ which has no pseudo limit in $V$. Then
$y\not\in K$ and so it is transcendental over $K$. Then
$V^{\prime\prime}=V^{\prime}\cap K(y)$ is a filtered union of localizations of
polynomial $V$-subalgebras in one variable by [13, Lemma 3.2] (see also [16,
Lemma 15]) and $V^{\prime}$ is a filtered union of its complete intersection
$V^{\prime\prime}$-subalgebras by Proposition 14, which is enough.

######  Theorem 16.

Let $V^{\prime}$ be an immediate extension of a valuation ring $V$ and
$K\subset K^{\prime}$ the fraction field extension. Then $V^{\prime}$ is a
filtered union of its complete intersection $V$-subalgebras.

###### Proof.

Firstly assume $K^{\prime}/K$ is of finite type, let us say
$K^{\prime}=K(x_{1},\ldots,x_{n})$. Set $V_{i}=V^{\prime}\cap
K(x_{1},\ldots,x_{i})$, $1\leq i<n$ and $V_{0}=V$, $V_{n}=V^{\prime}$. Then
$V_{i+1}$ is a filtered union of its complete intersection $V_{i}$-algebras
for all $0\leq i<n$ by Proposition 15 and we are done. In general, express
$K^{\prime}$ as a filtered union of some subfields $(K_{i})_{i}$ of
$K^{\prime}$ which are finite type extensions of $K$. Then $V^{\prime}$ is a
filtered increasing union of $V^{\prime}\cap K_{i}$.

######  Corollary 17.

Let $V$ be a valuation ring containing its residue field $k$ with a finitely
generated value group $\Gamma$. Then $V$ is a filtered union of its complete
intersection $k$-subalgebras. Moreover, if $F$ is the prime field contained in
$k$ then $V$ is a filtered union of its complete intersection $F$-subalgebras.

###### Proof.

By [16, Lemma 27 (1)] we see that $V$ is an immediate extension of a valuation
ring $V_{0}$ which is a filtered union of localizations of smooth $k$-algebras
(see [4, Theorem 1, in VI (10.3)], or [16, Lemma 27 (1)]). In fact, important
here is that there exists a cross-section $s:\Gamma\rightarrow K^{*}$ (see
[23, Proposition 5.4]), since $\Gamma$ is finitely generated and $V_{0}=V\cap
k(s(\Gamma))$ (a cross-section of $V$ is a section in the category of abelian
groups of the valuation map $\mathrm{val}:K^{*}\rightarrow\Gamma$). More
precisely, $V_{0}=V\cap k(y)$ for some elements $y$ from $V$ such that
$\mathrm{val}(y)$ is a basis in $\Gamma$. Now it is enough to apply Theorem 16
for $V_{0}\subset V$.

######  Corollary 18.

Let $V$ be a valuation ring containing a perfect field $F$ of characteristic
$p>0$, $k$ its residue field, $\Gamma$ its value group and $K$ its fraction
field. Assumethat $\Gamma$ is finitely generated, $k/F$ is separably generated
and $V$ is Henselian. Then $V$ is a filtered union of its complete
intersection $F$-subalgebras.

###### Proof.

The conclusion goes from Corollary 17 if we show that $k\subset V$. A lifting
of $k$ to $V$ could be done when $V$ is Henselian and char $k=p=0$ (see [23,
Theorem 2.9] but the proof goes in the same way when $p>0$ and $k$ is
separably generated over $F$. In particular the lifting could be done when
$k/F$ is of finite type.

######  Corollary 19.

Let $V$ be a valuation ring containing a perfect field $F$ of characteristic
$p>0$, $k$ its residue field, $\Gamma$ its value group and $K$ its fraction
field. Assume that there exists a cross-section $s:\Gamma\rightarrow K^{*}$
and either $k\subset V$, or $k/F$ is separably generated and $V$ is Henselian.
Then $V$ is a filtered union of its complete intersection $F$-subalgebras.

The proof goes as in Corollaries 17, 18 with $V_{0}=V\cap k(s(\Gamma))$.

To extend these results we need some facts from Model Theory (see next
section) and the following two lemmas, the first being [16, Lemma 7] which is
an extension of [9, Proposition 3], and [15, Proposition 5].

######  Lemma 20.

For a commutative diagram of ring morphisms

$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b\,\mapsto\,a}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V}$that
factors as
follows$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V/a^{3}V}$$\textstyle{A^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A^{\prime}/a^{3}A^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

with $B$ finitely presented over $A$, a $b\in B$ that is standard over $A$
(this means a special element from the ideal $H_{B/A}$ defining the non smooth
locus of $B$ over $A$, for details see for example [16, Lemma 4]), and a
nonzerodivisor $a\in A^{\prime}$ that maps to a nonzerodivisor in $V$ that
lies in every maximal ideal of $V$, there is a smooth $A^{\prime}$-algebra $S$
such that the original diagram factors as follows:

$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V.}$$\textstyle{A^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

######  Lemma 21.

Let $V\subset V^{\prime}$ be an extension of valuation rings and
${\hat{V}}\subset{\hat{V}^{\prime}}$ their completion extension. Assume that
the field extension of $V\subset V^{\prime}$ is separable. Then $V^{\prime}$
is a filtered direct limit of complete intersection $V$-algebras if
${\hat{V}}^{\prime}$ is a filtered direct limit of complete intersection
${\hat{V}}$-algebras. Moreover, if the above fraction field extension is not
separable, $E$ is a finitely presented $V$-algebra and $w:E\rightarrow
V^{\prime}$ is a $V$-morphism such that $w(H_{E/V})V^{\prime}\not=0$ then $w$
factors through a complete intersection $V$-algebra if ${\hat{V}}^{\prime}$ is
a filtered direct limit of complete intersection ${\hat{V}}$-algebras.

###### Proof.

We follow the proofs of [16, Lemma 7, Proposition 9]. Assume that
${\hat{V}}^{\prime}$ is a filtered direct limit of complete intersection
$\hat{V}$-algebras. Let $E$ be a finitely generated $V$-subalgebra of
$V^{\prime}$ and $w$ the inclusion $E\subset V^{\prime}$. $E$ is finitely
presented by [11, Theorem 4]. By separability of the fraction field extension
of $V\subset E$, the singular locus $w(H_{E/V})V^{\prime}$ has the form
$zV^{\prime}$ for some nonzero $z\in V^{\prime}$. The same thing follows when
the separability is missing but $w(H_{E/V})V^{\prime}\not=0$ . Actually,
changing if necessary $E$ by $E[Z]$ and $w$ by the possible non injective map
$E[Z]\rightarrow V^{\prime}$ given by $Z\rightarrow z$ we may assume that
$w(H_{E/V})V^{\prime}$ has the form $dV^{\prime}$ for some $d\in E$.

Suppose that $E\cong V[Y]/I$, $Y=(Y_{1}\ldots,Y_{m})$, $I$ being a finitely
generated ideal. As in the proof of [16, Proposition 20] we may assume that
for some polynomials $f=(f_{1},\ldots,f_{r})$ from $I$, we have $d\in NME$ for
some $N\in((f):I)$ and a $r\times r$-minor $M$ of the Jacobian matrix
$(\partial f_{i}/\partial Y_{j})$.

Note that ${\hat{w}}$ the composite map
$E\xrightarrow{w}V^{\prime}\subset{\hat{V}}^{\prime}$ factors through a
complete intersection $V$-algebra $B$, let us say $\hat{w}$ is the composite
map $E\rightarrow B\xrightarrow{\rho^{\prime}}{\hat{V}}^{\prime}$. We may
assume that $B$ has the form $T_{h}/(P)$ for a polynomial $V$-algebra
$T=V[U]$, $U=(U_{1},\ldots,U_{l})$, $h\in T$ and a regular system of elements
$P=(P_{1},\ldots,P_{t})$ from $T$. Suppose the map $\rho^{\prime}$ is given by
$U\rightarrow{\hat{u}}$ for some ${\hat{u}}\in({\hat{V}}^{\prime})^{l}$. Thus
$h({\hat{u}})$ is invertible. Clearly, the map $\bar{w}$ induced by $w$ modulo
$d^{3}$ factors through ${\bar{B}}=B/(d^{3})$, that is $\bar{w}$ is the
composite map
${\bar{E}}=E/(d^{3})\rightarrow{\bar{B}}\rightarrow{\bar{V}}=V/(d^{3})$, the
last map being given by $U\rightarrow u+d^{3}({\hat{V}}^{\prime})^{l}$, for
some $u\in(V^{\prime})^{l}$ which is $\hat{u}$ modulo $d^{3}$. We have
$P(u)=d^{3}z$ for some $z\in{\hat{V}}^{\prime t}$. Set
$D=(V[U,Z]/(P-d^{3}Z))_{h}$, $Z=(Z_{1},\ldots,Z_{t})$ and let
$\rho:D\rightarrow V$ be the map $(U,Z)\rightarrow(u,z)$. Note that $D$ is a
complete intersection $V$-algebra. As $w$ factors through $\rho$ modulo
$d^{3}$ there exists a smooth $D$-algebra $D^{\prime}$ such that $w$ factors
through $D^{\prime}$ using Lemma 20. It follows that $D^{\prime}$ is a
complete intersection $V$-algebra as well and it is enough to apply [18, Lemma
1.5].

##  3\. Model Theory of valued fields

There are two problems two extend Corollaries 17, 18. These are too reduce to
the case:

i) when $V$ has a cross-section $s:\Gamma\rightarrow K^{*}$,

ii) when $V$ contains its residue field.

For i) we may use [23, Proposition 5.4, Lemma 7.9], or [3, 3.3.39, 3.3.40].
Actually, they follow from the following two facts as F. V. Kuhlmann explain
to us

  1. (1)

every $\aleph_{1}$-saturated valued field admits a cross-section (see [23,
Lemma 7.9]),

  2. (2)

every valued field (or more generally, every first order structure in a small
enough language) admits an elementary extension which is
$\aleph_{1}$-saturated; in present model theoretic algebra, this fact (which
holds with any cardinal number $\geq\aleph_{1}$) has widely replaced the use
of ultraproducts.

In [16, Theorem A.10] there exists the following possible variant of these
results.

######  Theorem 22.

For a valuation ring $V$ with value group $\Gamma$, there is a countable
sequence of ultrafilters ${\mathscr{U}}_{1},{\mathscr{U}}_{2},\dots$ on some
respective sets $U_{1},U_{2},\dots$ for which the valuation rings
$\\{V_{n}\\}_{n\geq 0}$ defined inductively by $V_{0}:=V$ and
$V_{n+1}:=\prod_{{\mathscr{U}}_{n+1}}V_{n}$ are such that the valuation ring

$\textstyle\widetilde{V}:=\varinjlim_{n\geq 0}V_{n}\quad\text{has a cross-
section}\quad\widetilde{s}:\widetilde{\Gamma}\rightarrow\widetilde{K}^{*},$

where $\widetilde{K}$ and $\widetilde{\Gamma}$ are the fraction field and the
value group of $\widetilde{V}$.

######  Theorem 23.

Let $V$ be a valuation ring containing a perfect field $F$ of characteristic
$p>0$, $k$ its residue field, $\Gamma$ its value group and $K$ its fraction
field. Assume that $k\subset V$. Then $V$ is a filtered direct limit of
complete intersection $F$-algebras.

###### Proof.

Certainly we may assume that $V$ is not a field, otherwise the conclusion is
trivial. Let $E$ be a finitely generated $F$-subalgebra (in fact finitely
presented by [11, Theorem 4]) of $V$. We will show that $w$ factors through a
complete intersection $V$-algebra. It is enough to replace $V$ by the
intersection of $V$ with the fraction field $K_{E}$ of $E$.

As $K_{E}$ is a finite type field extension of $F$ we may assume $\Gamma$
countable generated and $k$ countable generated over $F$ by [1, Corollary
5.2]. Consider $U_{i}$, $\mathcal{U}_{i}$, $\widetilde{V}$, $\widetilde{k}$,
$\widetilde{\Gamma}$, $\widetilde{s}$, $\widetilde{t}$ given by Theorem 22 for
$V$. Note that $\widetilde{k}\subset\widetilde{V}$ because $k\subset V$. As
usual $\widetilde{V}$ is an immediate extension of a valuation ring
$\widetilde{V}_{0}=\widetilde{V}\cap\widetilde{k}(\widetilde{s}(\widetilde{\Gamma}))$,
which is a filtered union of localizations of polynomial algebras over
$\widetilde{k}$. Using Corollary 19 we see that $\widetilde{V}$ is a filtered
union of its complete intersection $\widetilde{V}_{0}$-subalgebras. Then the
map $w^{\prime}:\widetilde{k}\otimes_{F}E\rightarrow\widetilde{V}$ factors
through a complete intersection $\widetilde{k}$-algebra and so through a
complete intersection $F$-algebra $D$ since $\widetilde{k}/F$ is separable. So
$w$ factors through $D$ too as in the proof of [13, Corollary 5.4] because all
finite systems of polynomial equations which have a solution in $\tilde{V}$
must have one in $V$. The proof ends applying [18, Lemma 1.5].

The problem from ii) is also hard. As we explain in the proof of Corollary 18
the lifting of the residue field $k$ of $V$ when $p>0$ could be done when $V$
is Henselian and $k/{\bf F}_{p}$ is of finite type. To glue the liftings of
finite type subfields of $k$ is another problem which is solved using Model
Theory. On the other hand our method uses completions, which are Henselian but
in dimension one.

We need [16, Proposition A.6] (useful also in the proof of Theorem 22), which
is obtained using [5, Theorem 6.1.4] and says in particular the following:

######  Proposition 24.

Let $V$ be a valuation ring with value group $\Gamma$ and $\tau=card\ \Gamma$.
Then there exists an ultrafilter $\mathcal{U}$ on a set $U$ with $card\
U=\tau$ such that any system of polynomial equations $(g_{i}((X_{j})_{j\in
J})_{i\in I}$ with $card\ I\leq\tau$ in variables $(X_{j})_{j\in J}$ with
coefficients in the ultrapower ${\tilde{V}}=\Pi_{\mathcal{U}}V$ has a solution
in ${\tilde{V}}$ if and only if all its finite subsystems have.

######  Theorem 25.

For a Henselian valuation ring $V$ with residue field $k$ of characteristic
$p>0$, there is a countable sequence of ultrafilters
${\mathscr{U}}_{1},{\mathscr{U}}_{2},\dots$ on some respective sets
$U_{1},U_{2},\dots$ for which the valuation rings $\\{V_{n}\\}_{n\geq 0}$
defined inductively by $V_{0}:=V$ and
$V_{n+1}:=\prod_{{\mathscr{U}}_{n+1}}V_{n}$ for $n\geq 1$ are such that for
the valuation ring $\widetilde{V}:=\varinjlim_{n\geq 0}V_{n}$ there exists a
lifting $\widetilde{t}:\widetilde{k}\rightarrow\widetilde{V}$, where
$\widetilde{k}$ is the residue field of $\widetilde{V}$.

###### Proof.

Let $\mathcal{E}$ be the family of all subfields $k^{\prime}$ of $k$ which are
field extensions of finite type over ${\bf F}_{p}$. For such subfield
$k^{\prime}$ there exists a lifting $t_{k^{\prime}}:k^{\prime}\rightarrow V$
as we have seen above since $k^{\prime}/{\bf F}_{p}$ is separable generated.
We have $k^{\prime}={\bf F}_{p}({\bar{y}}_{k^{\prime}})$ for a finite system
of elements ${\bar{y}}_{k^{\prime}}$ from $k$. Let $G_{k^{\prime}}$ be the
polynomial equations from ${\bf F}_{p}[Y]$ satisfied by
${\bar{y}}_{k^{\prime}}$ (certainly $G_{k^{\prime}}$ could contain just one
equation if we arrange properly ${\bar{y}}_{k^{\prime}}$ but it has no
importance for us). The existence of $t_{k^{\prime}}$ says that
$G_{k^{\prime}}$ has a solution $y_{k^{\prime}}$ in $V$. If
$k^{\prime}_{1}\subset k^{\prime}_{2}$ for some fields
$k^{\prime}_{i}\in\mathcal{E}$, $i=1,2$ then the elements
${\bar{y}}_{k^{\prime}_{1}},{\bar{y}}_{k^{\prime}_{2}}$ satisfy some
polynomial equations $G_{k^{\prime}_{1}k^{\prime}_{2}}$ from ${\bf F}_{p}[Y]$,
several $Y$ which include $G_{k^{\prime}_{i}}$ for $i=1,2$. Certainly the
existence of $t_{k^{\prime}_{2}}$ assure us that
$G_{k^{\prime}_{1}k^{\prime}_{2}}$ has a solution in $V$. Let $G$ be the union
of all possible $G_{k^{\prime}_{1}k^{\prime}_{2}}$ when
$k^{\prime}_{1},k^{\prime}_{2}$ runs in $\mathcal{E}$.

Let $U_{1}$ be a set with $card\ U_{1}>card\ k$ and $\mathcal{U}_{1}$ an
ultrafilter given by Proposition 24. Then $G_{k}$ has a solution in
$V_{1}=\prod_{{\mathscr{U}}_{1}}V$ because each finite subsystems of $G_{k}$
has already a solution in $V$ given by the lifting of some big enough
$k^{\prime}\in\mathcal{E}$. So we get a lifting $t_{k}$ of $k$ in $V_{1}$.

Repeating this procedure we find some sets $(U_{n})$ and some ultrafilters
$(\mathcal{U}_{n})$ and define $V_{n}$ and $k_{n}$, so that
$k_{n+1}\cong\prod_{{\mathscr{U}}_{n+1}}k_{n}$ and
$V_{n+1}=\prod_{{\mathscr{U}}_{n+1}}V_{n}$ with

$\textstyle\widetilde{k}\cong\varinjlim_{n\geq 0}k_{n}\ \ \mbox{and}\ \
\widetilde{V}\cong\varinjlim_{n\geq 0}V_{n}.$

The idea is to build ultrafilters ${\mathscr{U}}_{n}$ one by one in such a way
that a desired map

$\textstyle\widetilde{t}\colon\widetilde{k}\rightarrow\widetilde{V}\ \
\mbox{would be the limit of compatible maps }\ \ t_{n}\colon k_{n}\rightarrow
V_{n+1}.$

######  Remark 26.

The proofs of Theorems 22, 25 are similar and in fact we may find
$\widetilde{V}$ such that there exist a cross-section
$\widetilde{s}:\widetilde{\Gamma}\rightarrow\widetilde{K}^{*}$ and a lifting
$\widetilde{t}:\widetilde{k}\rightarrow\widetilde{V}$ in the same time.

######  Theorem 27.

Let $V$ be a valuation ring containing a perfect field $F$ of characteristic
$p>0$. Assume that either that $\dim V=1$ or $V$ is Henselian. Then $V$ is a
filtered direct limit of complete intersection $F$-algebras.

###### Proof.

Let $k$ be the residue field of $V$, and $\Gamma$ its value group. Let $E$ be
a finitely generated $F$-subalgebra of $V$ and $w:E\rightarrow V$ the
inclusion. Since $V$ is separable over $F$ we see that $w(H_{E/F})V\not=0$,
where $H_{E/F}$ is the ideal defining the non smooth locus of $E$ over $F$ as
in Lemma 20. We will show that $w$ factors through a complete intersection
$V$-algebra. It is enough to replace $V$ by the intersection of $V$ with the
fraction field $K_{E}$ of $E$. Consider $U_{i}$, $\mathcal{U}_{i}$,
$\widetilde{V}$, $\widetilde{k}$, $\widetilde{\Gamma}$, $\widetilde{s}$,
$\widetilde{t}$ given by Theorems 22, 25 for $V$, or the completion $\hat{V}$
of $V$ in case $\dim V=1$. Thus in both case $V$ is Henselian and so
$\widetilde{V}$ is Henselian too. Thus we may assume
$\widetilde{k}\subset\widetilde{V}$ and so it is enough to continue as in
Theorem 16. As usual $\widetilde{V}$ is an immediate extension of a valuation
ring
$\widetilde{V}_{0}=\widetilde{V}\cap\widetilde{k}(\widetilde{s}(\widetilde{\Gamma}))$,
which is a filtered union of localizations of polynomial algebras over
$\widetilde{k}$. Using Corollary 19 we see that $\widetilde{V}$ is a filtered
direct union of its complete intersection $\widetilde{V}_{0}$-subalgebras.
Then the map $w^{\prime}:\widetilde{k}\otimes_{F}E\rightarrow\widetilde{V}$
factors through a complete intersection $\widetilde{k}$-algebra and so through
a complete intersection $F$-algebra $D$ since $\widetilde{k}/F$ is separable.
So $w$ or the composite map $E\rightarrow V\rightarrow\hat{V}$ when $\dim V=1$
factors through $D$ too because all finite systems of polynomial equations
which have a solution in $\tilde{V}$ must have one in $V$ respectively
$\hat{V}$ when $\dim V=1$. As $w(H_{E/F})V\not=0$ we see that $w$ factors also
by $D$ when $\dim V=1$ using Lemma 21. The proof ends applying [18, Lemma
1.5].

## References

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http://stacks.math.columbia.edu.

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# First-order quantum correction to the ground-state energy density of two-
dimensional hard-sphere Bose atoms

Sang-Hoon Kima,b<EMAIL_ADDRESS>Mukunda P. Dasb<EMAIL_ADDRESS>a
Division of Liberal Arts $\&$ Sciences, Mokpo National Maritime University,
Mokpo 530-729, Republic of Korea
b Department of Theoretical Physics, RSPhysSE, Institute of Advanced Studies,
The Australian National University, Canberra, ACT 0200, Australia

###### Abstract

Divergence exponents of the first-order quantum correction of a two-
dimensional hard-sphere Bose atoms are obtained by an effective field theory
method. The first-order correction to the ground-state energy density with
respect to the zeroth-order is given by ${\cal E}_{1}/{\cal
E}_{0}\sim|D-2|^{-\alpha}|\ln\gamma|^{-\alpha^{\prime}}$, where $D$ is the
spatial dimension, and $\gamma$ is the gas parameter ($\gamma=na^{D}$). As
$D\rightarrow 2$, $\alpha=\alpha^{\prime}=1$. We show that the first-order
quantum correction of the energy density is not perturbative in low dimensions
of $D<2.2$ regardless of any gas parameter which is much less that 1.

Boson systems, interacting boson model, quantum field theory

###### pacs:

05.30.Jp, 21.60.Fw, 03.70.+k

##  I Introduction

Weakly interacting uniform Bose gas is a fundamental topic of many-body Bose
systems and has been studied for more than five decades. Bose-Einstein
condensation(BEC) is known to be the only phase transition that does not
require interaction. However, in the real system an interaction exists even in
very dilute Bose gases, and often the diluteness is explained by a
D-dimensional gas parameter: $\gamma(D)=na^{D}\ll 1$, where $n$ is the
D-dimensional density and $a$ is the s-wave scattering length. At zero
temperature the chemical potential, energy density, number density, and speed
of sound are written as an expansion in powers of the gas parameter $\gamma$.

A fundamental approximation method of a weakly interacting Bose system was
introduced by Bogoliubov in 1947 bogoliubov . In three-dimensions(3D) the
ground-state energy density is given by

$\small{\cal E}_{0}+{\cal E}_{1}+\cdots={\cal
E}_{0}\left[1+c_{1}\sqrt{\gamma}+c_{2}\gamma\ln\gamma+\cdots\right].$ (1)

Similarly, the number density in 3D is given by

$\small
n_{0}+n_{1}+\cdots=n_{0}\left[1+\frac{8}{3\sqrt{\pi}}\sqrt{\gamma}+\cdots\right].$
(2)

${\cal E}_{0}=2\pi\hbar^{2}an^{2}/m$ is the zeroth-order or mean-field energy
density. The first-order quantum correction $c_{1}=128/15\sqrt{\pi}$ was first
obtained by Lee and Yang in 1957 for a hard-sphere Bose gas lee . Later, it
has been shown by Brueckner et al. brueckner ; beliaev  that above results are
generally true for any short-range potential with scattering length $a$. The
second-order quantum correction $c_{2}=8(4\pi-3\sqrt{3})/3\sqrt{\pi}$ that was
originated from a three-body interaction was obtained by Wu et al. in 1959 wu
; hugenholtz ; sawada . Recently, an effective field theory(EFT) method has
been suggested to obtain the coefficients(Braaten and Nieto braaten ). This
method was introduced by Georgi et al. in 1990s georgi ; kaplan ; manohar .

In two-dimensions(2D) an uniform Bose gas is governed by strong long-range
fluctuations. The fluctuations inhibit the formation of a true long-range
order. Therefore, 2D uniform Bose gas does not undergo BEC transition at
finite temperatures ketterle . However, this 2D system turns superfluid below
a certain temperature while under confinement. The ground-state energy density
of hard-sphere bosons in 2D has been studied by Schick in 1971 schick .

$\small{\cal E}_{0}+{\cal E}_{1}+\cdots={\cal E}_{0}\left[1+{{\cal
O}}(1/\ln\gamma)\right],$ (3)

and number density is

$\small
n_{0}+n_{1}+\cdots=n_{0}\left[1+\frac{1}{\ln(1/\gamma)}+\cdots\right].$ (4)

${\cal E}_{0}=2\pi\hbar^{2}n^{2}/(m|\ln\gamma|)$ is the mean-field energy
density. The first-order number density in 2D is finite as
$n_{1}/n_{0}=1/\ln(1/\gamma)\ll 1$, but the first-order quantum correction of
the ground-state energy density has not been obtained yet because an infrared
divergence of an integral prevents a direct calculation. It has the divergence
form of

$\displaystyle\left|\frac{{\cal E}_{1}}{{\cal E}_{0}}\right|$
$\displaystyle\sim$
$\displaystyle\frac{1}{|D-2|^{\alpha}|\ln\gamma|^{\alpha^{\prime}}},$ (5)

where the exponents $\alpha$ and $\alpha^{\prime}$ are positive constants.

In this paper, we will calculate the two exponents of divergence $\alpha$ and
$\alpha^{\prime}$ analytically. The EFT has an advantage that it works even in
non-integer dimensions. Therefore, we can rewrite the EFT in non-integer
dimensions between two and three. Then, taking the limit of
$D=\lim_{\epsilon\rightarrow o^{+}}2(1+\epsilon)$, we will obtain the two
exponents of divergence $\alpha$ and $\alpha^{\prime}$. Furthermore, since the
first-order quantum correction of the ground-state energy density converges in
D=3 but diverges in D=2, we may find the dimensional border of the
perturbative method which is effective between two and three.

The paper is organized as follows: In Section II, we review the basic
structure of EFT of a dilute Bose system in non-integer dimensions between two
and three braaten ; andersen ; rakimov . In Section III, we reproduce the
zeroth-order or mean-field results in D-dimensions. In section VI, we obtain
the first-order quantum correction in 2D and two exponents $\alpha$ and
$\alpha^{\prime}$. Also we interpolate the first-order quantum depletion
between 2D and 3D. Finally, in section V we summarize the results.

##  II Effective field theory by an effective Lagrangian

EFT is a general approach that can be used to analyze the low energy behavior
of a physical system and takes advantage of the separation of scales to make
model-independent predictions georgi ; kaplan ; manohar . The effective
lagrangian that describes the low energy physics is written in terms of only
the long-wavelength degrees of freedom that includes every non-renormalizable
interactions, but to a certain order in a low energy expansion, only a finite
number of operators contribute to a physical quantity andersen .

An uniform Bose gas can be described by field theoretic method with
Hamiltonian ${{\cal H}}-\mu N$, where $\mu$ is the chemical potential and $N$
is the number operator. The number of atoms are conserved by the phase
transformation: $\psi({\bf r},t)\rightarrow e^{i\theta}\psi({\bf r},t)$. The
free energy in ground-state of the system is

$\small{\cal F}(\mu)=\langle{{\cal H}}-\mu N\rangle_{\mu},$ (6)

where $\langle\cdots\rangle_{\mu}$ denotes the expectation value in the ground
state. Therefore, $\langle{\cal H}\rangle_{\mu}={\cal E}(\mu)$ and $\langle
N\rangle_{\mu}=n(\mu)$ are the energy and number in the ground state. The free
energy ${\cal F}$ is given by all connected vacuum diagrams that are Feynman
diagrams with no external legs braaten . The number density $n$ is given by
the expectation value of $(\psi^{\dagger}\psi)$ in the ground-state (we set
$\hbar=1$ for convenience)

$\displaystyle n(\mu)$ $\displaystyle=$
$\displaystyle\frac{1}{Z}\int{\cal{D}\psi^{\dagger}\cal{D}\psi(\psi^{\dagger}\psi)}\,e^{iS[\psi^{\dagger},\psi]}$
(7)

where $\psi^{\dagger}$ and $\psi$ are complex field operators of bosons, $S$
is the action from the Lagrangian braaten , and $Z$ is the grand-canonical
partition function given by

$\small
Z=\int{\cal{D}\psi^{\dagger}\cal{D}\psi}\,e^{iS[\psi^{\dagger},\psi]}.$ (8)

The grand-canonical ensemble of uniform Bose particles with a two-body
interaction is governed by the action

$\small S[\psi^{\dagger},\psi]=\int dt\int
d^{D}x\left[\psi^{\dagger}\left(i\frac{\partial}{\partial
t}+\frac{\mbox{\boldmath$\nabla$}^{2}}{2m}+\mu\right)\psi-\frac{1}{2}g\left(\psi^{\dagger}\psi\right)^{2}\right],$
(9)

where $m$ is the atomic mass, and $g$ is the D-dimensional coupling constant
which contains pairwise interaction between atoms. In D-dimensions $g$ is
given by nogueira

$\small
g(D)=\frac{4\pi^{D/2}a^{D-2}}{m}\frac{1}{2^{2-D}\Gamma(1-\frac{D}{2})\gamma^{D/2-1}+\Gamma(\frac{D}{2}-1)},$
(10)

where $\Gamma$ is the Gamma function. It satisfies the two limiting cases. In
3D, it has the well-know form

$\small g(3)=\frac{4\pi a}{m}.\\\ $ (11)

In 2D, it has the logarithmic form kim ; lieb

$\small g(2)=\frac{4\pi}{m}\frac{1}{\ln(1/\gamma)}.$ (12)

When the temperature of a Bose system falls below the condensation temperature
$T_{c}$, we can write the quantum field $\psi$ in terms of a time-independent
condensate $\phi$ and a quantum fluctuation field $\tilde{\psi}$

$\small\psi=\phi+\tilde{\psi}.$ (13)

The fluctuation field $\tilde{\psi}$ can be conveniently written in terms of
two real fields $\psi_{1}$ and $\psi_{2}$

$\small\tilde{\psi}=\frac{\psi_{1}+i\psi_{2}}{\sqrt{2}}.$ (14)

In the uniform system $\phi$ is the condensate order parameter and a real
constant. It corresponds to a breaking of the global $U(1)$ symmetry and takes
into account the Bogoluibov shift of the field operator.

The conservation of particle numbers requires that $\tilde{\psi}$ has non-zero
momentum component so that $\langle\psi\rangle=\phi$ and
$\langle\tilde{\psi}\rangle=0$. Therefore, the condensate order parameter
$\phi$ defines the density of condensed particles while $\tilde{\psi}$ defines
the density of uncondensed particles. Then, the zeroth- and first-order
quantum correction to the density are given by

$\small n_{0}=\phi^{2},\quad
n_{1}=\langle{\tilde{\psi}}^{\dagger}\tilde{\psi}\rangle.$ (15)

Substituting Eqs. (13) and (14) into Eq. (9) of the action $S$, we decompose
the real part of the action into three parts braaten ; andersen ; rakimov :

$\small
S[\phi,\psi_{1},\psi_{2}]=S_{clas}[\phi]+S_{free}[\phi,\psi_{1},\psi_{2}]+S_{int}[\phi,\psi_{1},\psi_{2}].$
(16)

$S_{clas}$ is the classical part of the action. It does not contain any filed
operator:

$\small S_{clas}[\phi]=\int dt\int
d^{D}x\left(\mu\phi^{2}-\frac{1}{2}g\phi^{4}\right).$ (17)

$S_{free}$ is the free part of the action. It is quadratic in $\psi_{1}$ and
$\psi_{2}$:

$\small S_{free}[\phi,\psi_{1},\psi_{2}]=\int dt\int
d^{D}x\left[\frac{1}{2}(\dot{\psi}_{1}\psi_{2}-\psi_{1}\dot{\psi}_{2})+\frac{1}{2}\psi_{1}\left(\frac{\mbox{\boldmath$\nabla$}^{2}}{2m}+X\right)\psi_{1}+\frac{1}{2}\psi_{2}\left(\frac{\mbox{\boldmath$\nabla$}^{2}}{2m}+Y\right)\psi_{2}\right],$
(18)

where $\dot{\psi}\equiv\partial\psi/\partial t$, and the two new variable $X$
and $Y$ are

$\displaystyle X$ $\displaystyle=$ $\displaystyle\mu-3g\phi^{2},$ (19)
$\displaystyle Y$ $\displaystyle=$ $\displaystyle\mu-g\phi^{2}.$ (20)

The terms $3g\phi^{2}$ and $g\phi^{2}$ in the $X$ and $Y$ are mean-field self-
energies of the system.

$S_{int}$ is the interaction part of the action. It is the remaining terms:

$\small S_{int}[\phi,\psi_{1},\psi_{2}]=\int dt\int
d^{D}x\left[\sqrt{2}J\psi_{1}+\frac{K}{\sqrt{2}}\psi_{1}(\psi_{1}^{2}+\psi_{2}^{2})-\frac{g}{8}(\psi_{1}^{2}+\psi_{2}^{2})^{2}\right],$
(21)

where

$\small J=\phi(\mu-g\phi^{2})=\phi Y,$ (22)

and

$\small K=-g\phi.$ (23)

The free part of the action in Eq. (18) gives rise to a propagator, which can
be used in perturbative framework. If we take the Fourier Transform in a
momentum space, the fluctuating part is given by

$\small\tilde{\psi}(t,{\bf
r})=\displaystyle\frac{1}{\sqrt{V}}\sum_{n=-\infty}^{\infty}\sum_{p}\tilde{\psi}(\omega_{n},{\bf
p})\exp\\{-i\omega_{n}t+i{\bf p}\cdot{\bf r}\\},$ (24)

where $V^{-1}\displaystyle{\sum_{p}}=\int d^{D}p/(2\pi)^{D}$, and
$\omega_{n}=2\pi nT$ is the Matsubara frequency. Therefore, the propagator of
the free action $S_{free}$ is written as

$\displaystyle
G(\omega,p)=\displaystyle\frac{1}{\omega^{2}-E_{p}^{2}+iE_{p}}\left(\begin{array}[]{cc}\frac{p^{2}}{2m}-Y&-i\omega\\\
i\omega&\frac{p^{2}}{2m}-X\end{array}\right),$ (27)

with the dispersion relation

$\small
E_{p}=\sqrt{\left(\frac{p^{2}}{2m}-X\right)\left(\frac{p^{2}}{2m}-Y\right)}.$
(28)

This is a general form of the dispersion and includes every information of the
energy spectrum in the two self-terms $X$ and $Y$. It is possible to
diagonalize the propagator matrix in Eq. (27) by a field redefinition or
renormalization, which is equivalent to the Bogoliubov transformation in the
operator method. However, such a redefinition makes the interaction terms in
the action more complicated and increases the number of diagrams. Therefore,
we use the propagator matrix with off-diagonal elements to minimize the number
of diagrams.

##  III Thermodynamic potential and zeroth-order results

It is convenient to introduce the thermodynamic potential $\Omega(\mu,\phi)$.
The thermodynamic potential contains the information required to determine all
of the thermodynamic functions. The free energy ${\cal F}(\mu)$ can be
obtained by evaluating $\Omega$ at a particular value of $\phi$.

The sum of the vacuum graphs is independent of the arbitrary background
condensate $\phi$. Thus, the sum of connected vacuum diagram reduces to the
sum of one-particle irreducible vacuum diagrams braaten ; andersen ; rakimov :

$\small{\cal F}(\mu)=\Omega(\mu,\phi(\mu))$ (29)

From Eq. (6) the ground-state energy density is written by

$\small{\cal E}(\mu)={\cal F}(\mu)+n\mu.$ (30)

By differentiating the free energy, we obtain the density and chemical
potential, too.

$\displaystyle n(\mu)=-\frac{d{\cal F}}{d\mu},\quad\mu(n)=\frac{d{\cal
E}}{dn}.$ (31)

The $n$-loop contribution to the $\Omega$ is denoted by $\Omega_{n}$ in FIG.
1. It is given by all one-particle irreducible vacuum diagrams and can be
expanded in the number of loops:

$\small\Omega(\mu,\phi)=\Omega_{0}(\mu,\phi)+\Omega_{1}(\mu,\phi)+\Omega_{2}(\mu,\phi)+\cdots.$
(32)

If $\Omega$ is evaluated at a value of the condensate, all one-particle
reducible diagrams vanish. Then, the free energy density in Eq. (29) is

$\small{\cal
F}(\mu)=\Omega_{0}(\mu,\phi)+\Omega_{1}(\mu,\phi)+\Omega_{2}(\mu,\phi)+\cdots.$
(33)

Figure 1:  Loop diagram of the two-body interaction. The zeroth, first, and
second loop from the left to right

Using Eqs. (13) and (14), the condition of the condensate reduces to
$\langle\psi_{1}\rangle=\langle\psi_{2}\rangle=0$. This condition is
equivalent to

$\small\frac{\partial\Omega(\mu,\phi)}{\partial\phi}=\frac{\partial\Omega_{0}(\mu,\phi)}{\partial\phi}+\frac{\partial\Omega_{1}(\mu,\phi)}{\partial\phi}+\frac{\partial\Omega_{2}(\mu,\phi)}{\partial\phi}+\cdots=0$
(34)

The number density in Eq. (31) is obtained from the thermodynamic potential,
too.

$\small n(\mu)=-\frac{\partial\Omega}{\partial\mu}.$ (35)

The free energy can also be expanded in powers of quantum corrections around
the mean-field value ${\cal F}_{0}(\mu)$:

$\small{\cal F}(\mu)={\cal F}_{0}(\mu)+{\cal F}_{1}(\mu)+{\cal
F}_{2}(\mu)+\cdots.$ (36)

The loop expansion in Eq. (33) does not coincide with the above expansion of
${\cal F}(\mu)$ in powers of quantum corrections because of its independence
of $\phi$. To obtain the expansion of ${\cal F}$ in powers of quantum
corrections, we must expand the condensate $\phi$ around its classical minimum
$\phi_{0}$, which satisfies

$\small\frac{\partial\Omega_{0}(\mu,\phi_{0})}{\partial\psi}=0.$ (37)

By expanding Eq. (34) in powers of $\phi-\phi_{0}$, and solving for $\phi$, we
obtain the quantum expansion for the condensate:

$\small\phi=\phi_{0}+\phi_{1}+\phi_{2}+\cdots,$ (38)

where $\phi_{n}$ is the $n$th-order quantum correction. For instance, the
first-order quantum correction for the condensate $\phi_{1}$ is obtained from
the expanding of Eq. (37) around $\phi_{0}$. The nonzero terms are

$\small\frac{\partial\Omega_{0}(\mu,\phi)}{\partial\phi}=\frac{\partial^{2}\Omega_{0}(\mu,\phi_{0})}{\partial\phi^{2}}(\phi-\phi_{0})+\frac{\partial\Omega_{1}(\mu,\phi_{0})}{\partial\phi}+\cdots=0.$
(39)

Then, $\phi_{1}$ in Eq. (38) is written as

$\small\phi_{1}=-\left[\frac{\partial\Omega_{1}(\mu,\phi_{0})}{\partial\phi}\right]\left[\frac{\partial^{2}\Omega_{0}(\mu,\phi_{0})}{\partial\phi^{2}}\right]^{-1}.$
(40)

Therefore, keeping up to the terms of the second order and
$\phi-\phi_{0}\approx\phi_{1}$, the first three terms of the free energy
density is

$\displaystyle{\cal F}_{0}(\mu)$ $\displaystyle=$
$\displaystyle\Omega_{0}(\mu,\phi_{0}),$ (41) $\displaystyle{\cal F}_{1}(\mu)$
$\displaystyle=$ $\displaystyle\Omega_{1}(\mu,\phi_{0}),$ (42)
$\displaystyle{\cal F}_{2}(\mu)$ $\displaystyle=$
$\displaystyle\Omega_{2}(\mu,\phi_{0})+\frac{\partial\Omega_{1}(\mu,\phi_{0})}{\partial\phi}\phi_{1}+\frac{1}{2}\frac{\partial^{2}\Omega_{0}(\mu,\phi_{0})}{\partial\phi^{2}}\phi_{1}^{2}.$
(43)

The mean-field thermodynamic potential is given by the terms in the classical
action in Eq. (17) as

$\small\Omega_{0}(\mu,\phi)=-\frac{1}{V}S_{clas}=-\mu\phi^{2}+\frac{1}{2}g\phi^{4}.$
(44)

From Eq. (37) we obtain the classical minimum $\phi_{0}$ as

$\small\phi_{0}=\sqrt{\frac{\mu}{g}}$ (45)

Therefore, the mean-field free energy density in D-dimensions is obtained from
Eq. (41)

$\small{\cal F}_{0}(\mu)=\Omega_{0}(\mu,\phi_{0})=-\frac{\mu^{2}}{2g}.$ (46)

The mean-field number density is obtained from Eq. (7) as

$\small n(\mu_{0})=-\frac{\partial{\cal
F}(\mu_{0})}{\partial\mu}=\frac{\mu_{0}}{g}$ (47)

The mean-field ground-state energy density in D-dimensions is obtained from
Eq. (30)

$\small{\cal
E}_{0}^{(D)}=-\frac{\mu_{0}^{2}}{2g}+n\mu_{0}=\frac{1}{2}g(D)n^{2}.$ (48)

In 3D, since $g(3)=4\pi a/m$ from Eq. (11), the mean-field energy density is

$\small{\cal E}_{0}^{(3)}=\frac{2\pi an^{2}}{m}.$ (49)

In 2D, since $g(2)=4\pi/m\ln(1/\gamma)$ from Eq. (12), the mean-field energy
density is

$\small{\cal E}_{0}^{(2)}=\frac{2\pi an^{2}}{m\ln(1/\gamma)}.$ (50)

Therefore, this method reproduced the well-known mean-field energy densities
in Eqs. (1) and (3) successfully.

##  IV First-order quantum correction

The first-order quantum correction of the ground-state energy is obtained in
the following way. Substituting Eq. (45) into Eqs. (19) and (20), we obtain
the two variables $X$ and $Y$ at the minimum of $\phi_{0}$.

$\displaystyle X$ $\displaystyle=$ $\displaystyle\mu-3g\phi_{0}^{2}=-2\mu,$
(51) $\displaystyle Y$ $\displaystyle=$ $\displaystyle\mu-g\phi_{0}^{2}=0.$
(52)

The propagator and the dispersion relation in Eqs. (27) and (28) becomes

$\displaystyle
G(\omega,p)=\displaystyle\frac{1}{\omega^{2}-E_{p}^{2}+iE_{p}}\left(\begin{array}[]{cc}\frac{p^{2}}{2m}&-i\omega\\\
i\omega&\frac{p^{2}}{2m}+2\mu\end{array}\right),$ (55)

and from Eq. (47)

$\small E_{p}=\sqrt{\frac{p^{2}}{2m}\left(\frac{p^{2}}{2m}+2\mu\right)}.$ (56)

These are the original Bogoliubov results. $E_{p}$ is gapless and is linear
for small wave-vectors. For large wave-vectors, the dispersion relation
becomes $E_{p}\simeq p^{2}/2m+2\mu\simeq p^{2}/2m+2g(D)n.$ Therefore, the
$2g(D)n$ represents the mean-field energy in D-dimensions due to interaction
with the condensed particles.

In the Bogoliubov approximation one makes a pair approximation to the
Hamiltonian by neglecting terms with three and four operators bogoliubov . On
the other hand, in the Beliaev approximation, one goes one step further by
calculating the leading quantum corrections to the quasi-particle spectrum
beliaev . This is done by including all one-loop diagrams.

The free energy can be written with the inverse of the propagator. From Eq.
(8)

$\small Z={\rm exp}\left\\{i\int dt\int d^{D}x\,{\rm
det}G^{-1}(\omega,p)\right\\}.$ (57)

Therefore,

$\small{\cal F}={\rm Tr}\ln G^{-1}(\omega,p).$ (58)

Note that $\ln\rm{det}{\bf A}={\rm Tr}\ln{\bf A}$ for any matrix ${\bf A}$.
The one-loop contribution to the thermodynamic potential in D-dimensions,
$\Omega_{1}=i\ln Z$, is obtained by the following way.

$\displaystyle\Omega_{1}(\mu,\phi)$ $\displaystyle=$
$\displaystyle-\frac{i}{2}\int\frac{d\omega}{2\pi}\int\frac{d^{D}p}{(2\pi)^{D}}\ln\det
G(\omega,p)^{-1}$ (59) $\displaystyle=$
$\displaystyle\frac{1}{2}\int\frac{d^{D}p}{(2\pi)^{D}}E_{p}.$

From Eq. (42), we have the free energy component

$\small{\cal
F}_{1}(\mu)=\Omega_{1}(\mu,\phi_{0})=\frac{1}{4m}I_{0,-1}(\kappa),$ (60)

where $\kappa=4m\mu=4mgn$ and $I_{i,j}$ are loop integral function in
$D$-dimensions, which is defined as braaten ; andersen

$\displaystyle I_{i,j}^{(D)}(\kappa)$ $\displaystyle=$
$\displaystyle\int\frac{d^{D}p}{(2\pi)^{D}}\frac{p^{2i}}{p^{j}(p^{2}+\kappa)^{j/2}},$
(61)

where $i$ and $j$ are integers. This $I_{i,J}$ is expressed by Gamma function,
too.

$\small
I_{i,j}^{(D)}=\frac{1}{(4\pi)^{D/2}}\frac{\Gamma\left(\frac{D+2i-j}{2}\right)\Gamma\left(-\frac{D+2i-2j}{2}\right)}{\Gamma\left(\frac{D}{2}\right)\Gamma\left(\frac{j}{2}\right)}\kappa^{\frac{D+2i-2j}{2}}.$
(62)

Note that in D=3, $I_{0,-1}^{(3)}(\kappa)=\kappa^{5/2}/15\pi^{2}$,
$I_{1,1}^{(3)}(\kappa)=\kappa^{3/2}/3\pi^{2}$, and
$I_{-1,-1}^{(3)}(\kappa)=-\kappa^{3/2}/6\pi^{2}$. It has the useful relation
of the first derivative

$\small\frac{d}{d\kappa}I_{i,j}^{(D)}(\kappa)=-\frac{j}{2}I_{i+1,j+1}^{(D)}(\kappa).$
(63)

Ground-state energy density is expressed as

$\displaystyle{\cal E}_{0}+{\cal
E}_{1}+\cdots=\frac{1}{2}gn^{2}+\frac{1}{4m}I^{(D)}_{0,-1}(\kappa)+\cdots.$
(64)

Therefore, the contribution of the first-order quantum correction in
D-dimensions is obtained as

$\displaystyle\frac{{\cal E}_{1}}{{\cal E}_{0}}$ $\displaystyle=$
$\displaystyle-\frac{\Gamma\left(-\frac{D+2}{2}\right)\Gamma\left(\frac{D+1}{2}\right)}{\pi^{(D+1)/2}\Gamma\left(\frac{D}{2}\right)}\left\\{mg(D)\right\\}^{D/2}n^{D/2-1}$
(65) $\displaystyle=$
$\displaystyle-\frac{2^{D}\pi^{(D^{2}-2D-2)/4}\Gamma\left(-\frac{D+2}{2}\right)\Gamma\left(\frac{D+1}{2}\right)\gamma^{D/2-1}}{\Gamma\left(\frac{D}{2}\right)\left\\{2^{2-D}\Gamma\left(1-\frac{D}{2}\right)\gamma^{D/2-1}+\Gamma\left(\frac{D}{2}-1\right)\right\\}^{D/2}}.$

Note that $\gamma=(n_{0}+n_{1}+\cdots)a^{D}$.

Figure 2:  The first-order quantum correction of the ground-state energy
density in D-dimensions in the logarithmic scale. The divergence in D=2 is
clear. $\gamma$ itself is a function of the dimensions.

Substituting D=3, we obtain the well-known first-order quantum correction of
the ground-state energy density in 3D as

$\small\frac{{\cal E}_{1}}{{\cal
E}_{0}}=\frac{128}{15\sqrt{\pi}}\sqrt{\gamma}.$ (66)

Taking the limit of $D=\lim_{\varepsilon\to 0^{+}}2(1+\varepsilon)$, we obtain
the first-order quantum correction of the ground-state energy density in 2D as

$\small\frac{{\cal E}_{1}}{{\cal
E}_{0}}\simeq\frac{\gamma^{\varepsilon}}{-\gamma^{\varepsilon}+1}\simeq\frac{1}{\varepsilon\ln(1/\gamma)}\gg
1.$ (67)

For the singularities of the $\Gamma$ function at non-positive argument, we
used the following expansions

$\small\Gamma(-n+\varepsilon)=\frac{(-1)^{n}}{n}\left[\frac{1}{\varepsilon}+h(n+1)+\cal{O}(\varepsilon)\right],$
(68)

where $h(n+1)=1+1/2+1/3+\cdots+1/n-\gamma_{E}$ and $\gamma_{E}=0.5772...$ is
the Euler constant. In particular,
$\Gamma(\varepsilon)=1/\varepsilon-\gamma_{E}$. Also, we used the
approximations: $\gamma^{\varepsilon}\simeq 1+\varepsilon\ln\gamma$.

In 2D there is a divergence, and we can obtain the two exponents of the
divergence in Eq. (5) as $\alpha=\alpha^{\prime}=1$. We plotted ${\cal
E}_{1}/{\cal E}_{0}$ in FIG. 2 as the functions of the dimensions and gas
parameter in the logarithmic scale. From the FIG. 2 it is clear that the
expansion in Eq. (3) satisfy the perturbative condition of ${\cal E}_{1}/{\cal
E}_{0}\ll 1$ only when $D>2.2$. In low dimensions of $D<2.2$, the first-order
ground-state energy density is not perturbative regardless of the magnitude of
the gas parameter.

The number density is obtained from Eqs. (13)-(15) as

$\small n=\phi^{2}+\frac{1}{2}\langle\psi_{1}^{2}+\psi_{2}^{2}\rangle.$ (69)

Therefore, taking the one-loop effects into account

$\displaystyle n_{0}+n_{1}$ $\displaystyle=$ $\displaystyle
n_{0}+\frac{i}{2}\int\frac{d\omega}{2\pi}\frac{d^{D}p}{(2\pi)^{D}}\left[\frac{p^{2}/2m+2mE_{p}^{2}/p^{2}}{\omega^{2}-E_{p}^{2}+iE_{p}}\right]$
(70) $\displaystyle=$ $\displaystyle
n_{0}+\frac{1}{4}I_{1,1}(4mgn)+\frac{1}{4}I_{-1,-1}(4mgn)$ $\displaystyle=$
$\displaystyle
n_{0}\left[1+\frac{\Gamma\left(-\frac{D}{2}\right)\left\\{2\Gamma\left(\frac{D+1}{2}\right)-\Gamma\left(\frac{D-1}{2}\right)\right\\}}{2^{3}\pi^{(D+2)/2}\Gamma\left(\frac{D}{2}\right)}(mg)^{D/2}n_{0}^{(D-2)/2}\right].$

Then, the first-order quantum correction of the number density in D-dimensions
is obtained as

$\displaystyle\frac{n_{1}}{n_{0}}$ $\displaystyle=$
$\displaystyle\frac{\pi^{(D^{2}-2D-2)/4}}{2^{3-D}}\frac{\Gamma\left(-\frac{D}{2}\right)\left\\{2\Gamma\left(\frac{D+1}{2}\right)-\Gamma\left(\frac{D-1}{2}\right)\right\\}}{\Gamma\left(\frac{D}{2}\right)\left\\{2^{2-D}\Gamma\left(1-\frac{D}{2}\right)\gamma^{D/2-1}+\Gamma\left(\frac{D}{2}-1\right)\right\\}^{D/2}}\gamma^{D/2-1}.$
(71)

In 3D we obtain the well-known first-order quantum depletion as

$\small\frac{n_{1}}{n_{0}}=\frac{8}{3\sqrt{\pi}}\sqrt{\gamma}.$ (72)

In 2D taking the limit of $D=\lim_{\varepsilon\to 0^{+}}2(1+\varepsilon)$, we
obtain the first-order quantum depletion as

$\displaystyle\frac{n_{1}}{n_{0}}$ $\displaystyle\simeq$
$\displaystyle\frac{\gamma^{\varepsilon}}{-\frac{1}{\varepsilon}\gamma^{\varepsilon}+\frac{1}{\varepsilon}}.$
(73) $\displaystyle\simeq$ $\displaystyle\frac{1}{\ln(1/\gamma)}.$

We used the approximations: $\Gamma(-\frac{D}{2})\sim 1/\varepsilon$ and
$2\Gamma(\frac{D+1}{2})-\Gamma(\frac{D-1}{2})\sim 2\sqrt{\pi}\varepsilon$ .
This result is the same as the Schick’s value in Eq. (4). We plotted
$n_{1}/n_{0}$ as a function of dimensions and gas parameter in Fig. 3. It is
increasing monotonically as the dimension is decreasing from 3 to 2. The
depletion is stronger in low dimensions.

Figure 3:  The first-order quantum correction of the number density at
$\gamma=10^{-4}$ as a function the dimensions.

##  V Summary

EFT is a theory of symmetries. Once the symmetries have been identified, one
writes down the most general local effective Lagrangian consistent with these
symmetries. At zero temperature, the symmetries are Galilean invariance, time-
reversal symmetries, and the global phase symmetry. These symmetries restricts
the possible terms in the effective action.

The first-order quantum correction to the ground-state energy density of a 2D
uniform Bose atoms has the form of the divergence: ${\cal E}_{1}/{\cal
E}_{0}\sim|D-2|^{-\alpha}|\ln\gamma|^{-\alpha^{\prime}}$. Applying an
effective field theory method to the hard-sphere boson system in D-dimensions,
we obtained the D-dimensional free energy and thermodynamic potential up to
the one-loop result. This general form in D-dimensions reproduced every well-
known result in 3D and 2D.

Then, taking the limit of $D=\lim_{\varepsilon\to 0^{+}}2(1+\varepsilon)$, we
obtained the zeroth- and first-order quantum correction in 2D. Finally, we
obtained the two exponents of divergence $\alpha$ and $\alpha^{\prime}$
analytically as $\alpha=\alpha^{\prime}=1$. We also showed that the
perturbative expression of the ground-state energy density is effective only
in low dimensions of $D<2.2$.

###### Acknowledgements.

This work is supported by Korea Science and Engineering Foundation(KOSEF) and
Australian Academy of Science(AAS) scientific exchange program in 2008.

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