# Toric and tropical Bertini theorems in positive characteristic

Francesca Gandini  ,  Milena Hering  ,  Diane Maclagan  ,  Fatemeh Mohammadi
,  Jenna Rajchgot  ,  Ashley K. Wheeler  and  Josephine Yu

###### Abstract.

We generalize the toric Bertini theorem of Fuchs, Mantova, and Zannier [FMZ]
to positive characteristic. A key part of the proof is a new algebraically
closed field containing the field $\mathbbm{k}(t_{1},\dots,t_{d})$ of rational
functions over an algebraically closed field $\mathbbm{k}$ of prime
characteristic. As a corollary, we extend the tropical Bertini theorem of
Maclagan and Yu [maclagan2019higher] to arbitrary characteristic, which
removes the characteristic dependence from the $d$-connectivity result for
tropical varieties from that paper.

##  1\. Introduction

Bertini’s theorem, which states that a general hyperplane section of an
irreducible variety is again irreducible, is a basic result in algebraic
geometry. This has been generalized in many different ways, most notably for
this paper by Fuchs, Mantova, and Zannier [FMZ], who replace hyperplane
sections by certain subtori when the variety is a subvariety of an algebraic
torus in characteristic zero. Our main result removes this characteristic
assumption, at the expense of some precision.

######  Theorem 1.1 (Toric Bertini).

Let $\mathbbm{k}$ be an algebraically closed field of arbitrary
characteristic. Let $X$ be a $d$-dimensional irreducible subvariety of
$(\mathbbm{k}^{*})^{n}$ with $d\geq 2$, and let
$\pi\colon(\mathbbm{k}^{*})^{n}\rightarrow(\mathbbm{k}^{*})^{d}$ be a morphism
with $\pi|_{X}$ dominant and finite. Suppose that the pullback of $\pi|_{X}$
along any isogeny
$\mu\colon(\mathbbm{k}^{*})^{d}\rightarrow(\mathbbm{k}^{*})^{d}$ is
irreducible. Then for every $1\leq r\leq d-1$ the set of $r$-dimensional
subtori $T\subseteq(\mathbbm{k}^{*})^{d}$ with
$\pi^{-1}(\boldsymbol{\theta}\cdot T)\cap X$ irreducible for all
$\boldsymbol{\theta}\in(\mathbbm{k}^{*})^{d}$ is dense in the Grassmannian
$\operatorname{Gr}(r,d)$.

In characteristic zero [FMZ] show that the conclusion holds for subtori $T$ in
a generic (Zariski open) set, rather than just in a dense set. More precisely,
there are a finite number of exceptional subtori that $T$ must avoid, and any
other $T$ will have irreducible preimage. A similar “generic” conclusion may
hold in characteristic $p$; we are not aware of a counterexample, though our
current techniques are insufficient for a proof. The techniques of [FMZ]
cannot be extended to characteristic $p$ as they rely on results that are
false in characteristic $p$.

The condition on the irreducibility of pullbacks along isogenies appears
already in [FMZ] and is necessary for a generic result; for example, for the
projection
$\pi:\operatorname{V}(x-y^{2}z^{2})\rightarrow(\mathbbm{k}^{*})^{2}$ onto the
first two coordinates, the preimage of any subtorus of $(\mathbbm{k}^{*})^{2}$
of the form $(t^{2a},t^{b})$ is reducible. Thus the set of desired subtori
cannot be open. However the denseness conclusion may still hold without this
pullback hypothesis. We do not have a counterexample; see Remark 4.8.

One consequence of Theorem 1.1 is that we can remove the characteristic
assumption in the Tropical Bertini theorem of Maclagan and Yu
[maclagan2019higher].

######  Theorem 1.2 (Tropical Bertini).

Let $X\subset(\mathbbm{k}^{*})^{n}$ be an irreducible $d$-dimensional variety,
with $d\geq 2$, over an algebraically closed valued field $\mathbbm{k}$ with
$\mathbb{Q}$ contained in the value group. The set of rational affine
hyperplanes $H$ in $\mathbb{R}^{n}$ for which the intersection
$\operatorname{trop}(X)\cap H$ is the tropicalization of an irreducible
variety is dense in the Euclidean topology on $\mathbb{P}^{n}_{\mathbb{Q}}$.

The characteristic zero case of the Tropical Bertini theorem was originally
introduced in [maclagan2019higher] to prove a higher connectivity result for
tropicalizations of irreducible varieties [maclagan2019higher]*Theorem 1.
Theorem 1.2 removes the characteristic assumption from that connectivity
theorem.

The key ingredient in the proof of Theorem 1.1 is a new field $K_{\mathbf{w}}$
that contains the algebraic closure of $\mathbbm{k}(t_{1},\dots,t_{d})$. This
field is smaller, in some crucial aspects, than previously constructed
algebraically closed fields containing $\mathbbm{k}(t_{1},\dots,t_{d})$ when
$\mathbbm{k}$ has characteristic $p$.

The use for this field is best illustrated by the case when the variety $X$ of
Theorem 1.1 is a hypersurface, the map $\pi\colon
X\rightarrow(\mathbbm{k}^{*})^{d}$ is projection onto the first $d$
coordinates, and the subtorus $T\subseteq(\mathbbm{k}^{*})^{d}$ is one-
dimensional: $T=\\{(t^{n_{1}},\dots,t^{n_{d}}):t\in\mathbbm{k}^{*}\\}$ for
some $\mathbf{n}=(n_{1},\dots,n_{d})\in\mathbb{Z}^{d}$. This is, in fact, the
core case of the proof. Given an irreducible $f\in\mathbbm{k}[t_{1}^{\pm
1},\dots,t_{d}^{\pm 1},y]$, we wish to show that the set of
$\mathbf{n}\in\mathbb{Z}^{d}$, for which the substitution
$g\in\mathbbm{k}[x^{\pm 1},y]$ given by

$g(x,y)=f(x^{n_{1}},\dots,x^{n_{d}},y)$

remains irreducible, is dense in $\mathbb{P}^{d-1}_{\mathbb{Q}}$. The key idea
is to regard $f$ as a polynomial in $y$ with coefficients in
$\mathbbm{k}(t_{1},\dots,t_{d})$. The polynomial $f$ then factors as

$f=\prod_{i=1}^{s}(y-\alpha_{i}),$

where $\alpha_{i}$ are in the algebraic closure of
$\mathbbm{k}(t_{1},\dots,t_{d})$. A precise description of the algebraic
closure is still unknown but several fields containing it are known; see
[Christol, KedlayaEarly, KedlayaErratum, AdamczewskiBell, McD, GonzalezPerez,
ArocaIlardi, ArocaRond, Saavedra]. When $d=1$ and
$\operatorname{char}(\mathbbm{k})=0$, the algebraic closure of
$\mathbbm{k}(t_{1})$ is contained in the field of Puiseux series
$\mathbbm{k}\\{\\!\\{t_{1}\\}\\!\\}=\bigcup_{n\geq
1}\mathbbm{k}(\\!(t_{1}^{1/n})\\!)$. The exponents appearing in a particular
Puiseux series all have a fixed common denominator. In characteristic $p$ this
is relaxed to allow arbitrary powers of $p$ in the denominator, subject to the
requirement, which is automatic for Puiseux series, that the set of all
exponents is well ordered. As we recall in Section 2, there are also
multivariate generalizations of this, so we may regard the roots $\alpha_{i}$
of $f$ as multivariate Laurent series with fractional exponents.

Given this description of the roots of $f$, the natural expectation is that
the roots of $g$ as a polynomial in $y$ are obtained from the series
$\alpha_{i}$ by the same specialization $t_{i}=x^{n_{i}}$. To show that $g$
remains irreducible for most $\mathbf{n}$ it then suffices to show that none
of these specializations of $\alpha_{i}$s, or the elementary symmetric
polynomials in them, which are the coefficients of any factors of $g$, are
polynomials in $x$, as opposed to generalized Puiseux series.

However it is far from clear that this specialization map is well defined. For
example, given the multivariate series

$\alpha=\sum_{j\geq 1}t_{1}^{1-1/2^{j}}t_{2}^{1/2^{j}}$

over a field of characteristic $2$, we cannot make the substitution
$t_{1}=t_{2}=x$. A contribution of this paper is to define a field
$K_{\mathbf{w}}$ that contains the algebraic closure of
$\mathbbm{k}(t_{1},\dots,t_{d})$, and has natural subrings on which this
specialization map is a well-defined ring homomorphism, so this plan of attack
goes through.

The structure of the paper is as follows. The new field $K_{\mathbf{w}}$ is
introduced in Section 2, and it is shown to be algebraically closed in Section
3. Theorem 1.1 is proved in Section 4, while Theorem 1.2 is proved in Section
5.

Acknowledgements. This project began at the BIRS workshop on Women in
Commutative Algebra, and we are grateful to the organizers for bringing us
together, and to BIRS for hosting. Maclagan was partially supported by EPSRC
grant EP/R02300X/1. Mohammadi was partially supported by EPSRC Early Career
Fellowship EP/R023379/1, UGent BOF/STA/201909/038, and FWO grants (G023721N,
G0F5921N). Rajchgot was partially supported by NSERC Grant RGPIN-2017-05732.
Yu was partially supported by NSF-DMS grant #1855726.

Notation. Throughout this paper, by a cone in $\mathbb{R}^{d}$ we mean a
convex set closed under multiplication by positive scalars. It is pointed if
its closure does not contain a line. For a cone $C$ in $\mathbb{R}^{d}$, the
dual cone $C^{\vee}$ is
$\\{\mathbf{x}\in\mathbb{R}^{d}:\mathbf{x}\cdot\mathbf{y}\geq 0\text{ for all
}\mathbf{y}\in C\\}$. We denote by $\operatorname{int}(C)$ the interior of a
cone $C$, and by $\mathbb{R}^{d}_{li}$ elements of $\mathbb{R}^{d}$ whose
coordinates are linearly independent over $\mathbb{Q}$.

##  2\. Field Families and $p$-discreteness

In this section we construct the field $K_{\mathbf{w}}$ containing
$\mathbbm{k}(t_{1},\dots,t_{d})$ that plays a key role in the proof of Theorem
1.1. We will prove in Section 3 that $K_{\mathbf{w}}$ is algebraically closed.

###  2.1. Field families

We first recall previous constructions of algebraically closed fields
containing $\mathbbm{k}(t_{1},\dots,t_{d})$.

When $\operatorname{char}(\mathbbm{k})=0$ and $d=1$, the field of Puiseux
series $\mathbbm{k}\\{\\!\\{t\\}\\!\\}:=\bigcup_{n\geq
0}\mathbbm{k}(\\!(t^{1/n})\\!)$ is algebraically closed and contains
$\mathbbm{k}(t)$. Puiseux series are not algebraically closed when
$\operatorname{char}(\mathbbm{k})=p>0$. This was first observed by Chevalley
[Chevalley], who observed that the Artin–Schreier polynomial $x^{p}-x-t^{-1}$
has no Puiseux series root. Abhyankar [abhyankar] showed that the series
$\sum_{j\geq 0}t^{1/p^{j}}$ is a root. This was generalized by Rayner
[Rayner1], who showed that the collection

(1) $\left\\{\sum c_{\alpha}t^{\alpha}:\\{\alpha:c_{\alpha}\neq 0\\}\text{ is
well ordered, and there is }N>0\text{ with }\alpha\in\bigcup_{j\geq
0}\frac{1}{Np^{j}}\mathbb{Z}\text{ for all }\alpha\right\\}$

forms an algebraically closed field. Rayner introduced the idea of a field
family, which gives conditions on the possibilities of supports for power
series of this form. We now recall this in our setting.

######  Definition 2.1.

Let $\mathcal{A}$ be a family of subsets of an ordered abelian group
$(\Gamma,\preceq)$. For $A\in\mathcal{A}$, let $S(A)$ denote the semigroup
generated by $A$ under addition in $\Gamma$. Then $\mathcal{A}$ is called a
_field family_ with respect to $\Gamma$ if the following axioms hold:

  1. (i)

Every $A\in\mathcal{A}$ is well ordered, i.e., every subset has a least
element with respect to $\preceq$.

  2. (ii)

The elements of members of $\mathcal{A}$ generate $\Gamma$.

  3. (iii)

If $A,B\in\mathcal{A}$ then $A\cup B\in\mathcal{A}$.

  4. (iv)

If $A\in\mathcal{A}$ and $B\subseteq A$ then $B\in\mathcal{A}$.

  5. (v)

Given $A\in\mathcal{A}$ and $\gamma\in\Gamma$, then $\gamma+A:=\\{\gamma+a:\
a\in A\\}\in\mathcal{A}$.

  6. (vi)

If $A\in\mathcal{A}$, with $a\succeq 0$ for all $a\in A$, then
$S(A)\in\mathcal{A}$.

Given any field $\mathbbm{k}$, let $\mathbbm{k}^{\Gamma}$ denote the set of
all mappings $\varphi:\Gamma\to\mathbbm{k}$. The _support_ of
$\varphi\in\mathbbm{k}^{\Gamma}$ is the set
$\operatorname{supp}{\varphi}=\\{\gamma\in\Gamma:\ \varphi(\gamma)\neq 0\\}$.
For a family $\mathcal{A}$ of subsets of $\Gamma$ define

$\mathbbm{k}^{\Gamma}(\mathcal{A})=\\{\varphi\in\mathbbm{k}^{\Gamma}:\
\operatorname{supp}{\varphi}\in\mathcal{A}\\}.$

We think of elements in $\mathbbm{k}^{\Gamma}(\mathcal{A})$ as formal power
series,

$\varphi=\sum_{\gamma\in\Gamma}\varphi(\gamma)\mathbf{t}^{\gamma}\in\mathbbm{k}^{\Gamma}(\mathcal{A}).$

For $\varphi,\psi\in\mathbbm{k}^{\Gamma}(\mathcal{A})$ and $\gamma\in\Gamma$,
addition is defined as

$(\varphi+\psi)(\gamma)=\varphi(\gamma)+\psi(\gamma)$

and multiplication is defined as

$(\varphi\cdot\psi)(\gamma)=\sum_{\gamma_{1}+\gamma_{2}=\gamma}\varphi(\gamma_{1})\psi(\gamma_{2}).$

######  Theorem ([Rayner1]*Theorem 1).

If $\mathcal{A}$ is a field family then $\mathbbm{k}^{\Gamma}(\mathcal{A})$ is
a field.

The field in (1) is the case when $\Gamma=\mathbb{Q}$ and $\mathcal{A}$
consists of all well-ordered subsets of $\mathbb{Q}$ that lie in
$\bigcup_{j\geq 0}\frac{1}{Np^{j}}\mathbb{Z}$ for some $N\in\mathbb{N}$.

When $\Gamma=\mathbb{Q}^{d}$ the field elements can be regarded as generalized
Puiseux series in $d$ variables $t_{1},\dots,t_{d}$. For
$\operatorname{char}(\mathbbm{k})=0$, in [McD] McDonald constructed
multivariate Puiseux series solutions $g$ for equations of the form
$f(t_{1},\dots,t_{d},g)=0$ where $f\in\mathbbm{k}[t_{1},\dots,t_{d},y]$. The
supports of the solution series are contained in sets of the form
$\frac{1}{N}\mathbb{Z}^{d}\cap C$ where $N\in\mathbb{N}$ and $C$ is a pointed
cone.

Using Rayner’s field family formalism Aroca and Ilardi constructed
algebraically closed fields containing $\mathbbm{k}(t_{1},\dots,t_{d})$ and
McDonald’s series [ArocaIlardi]. For $\operatorname{char}(\mathbbm{k})=p>0$
Saavedra gave an analogous construction of algebraically closed fields
containing $\mathbbm{k}(t_{1},\dots,t_{d})$ [Saavedra]. The supports of
Saavedra’s series are contained in sets of the form $\bigcup_{j\geq
0}\frac{1}{Np^{j}}\mathbb{Z}^{d}\cap C$ where $N\in\mathbb{N}$ and $C$ is a
pointed cone.

All of these fields contain elements that are not algebraic over
$\mathbbm{k}(t_{1},\dots,t_{d})$. While there is some work on characterizing
the algebraic closure of $\mathbbm{k}(t_{1},\dots,t_{d})$ [AdamczewskiBell,
ArocaRond, Christol, KedlayaEarly, KedlayaErratum] there is currently no
complete answer in multiple variables and arbitrary characteristic.

###  2.2. The field of $p$-discrete series

We now use the formalism of Definition 2.1 to construct a field containing
$\mathbbm{k}(t_{1},\dots,t_{d})$ but contained in Saavedra’s field. Let
$\mathbb{R}^{d}_{li}$ denote the set of vectors in $\mathbb{R}^{d}$ whose
coordinates are linearly independent over $\mathbb{Q}$. For
$\mathbf{w}\in\mathbb{R}^{d}_{li}$ we define an ordering
$\preceq_{\mathbf{w}}$ on the abelian group $\mathbb{Q}^{d}$ as follows:

$\mathbf{u}\preceq_{\mathbf{w}}\mathbf{v}\quad\text{ when
}\quad\mathbf{w}\cdot\mathbf{u}\leq\mathbf{w}\cdot\mathbf{v}.$

The condition that the entries of $\mathbf{w}$ are linearly independent over
the rationals ensures that $\preceq_{\mathbf{w}}$ is a total order.

######  Definition 2.2 ($p$-discreteness).

For a subset $A\subseteq\mathbb{Q}^{d}$,
$\boldsymbol{\gamma}\in\mathbb{Q}^{d}$, and
$\mathbf{w}\in\mathbb{R}^{d}_{li}$, set

$A^{+}_{\boldsymbol{\gamma},\mathbf{w}}=\\{\mathbf{a}\in
A:\mathbf{w}\cdot\mathbf{a}>\mathbf{w}\cdot\boldsymbol{\gamma}\\}\ \
\text{and}\ \ A^{-}_{\boldsymbol{\gamma},\mathbf{w}}=\\{\mathbf{a}\in
A:\mathbf{w}\cdot\mathbf{a}<\mathbf{w}\cdot\boldsymbol{\gamma}\\}.$

We say that a set $A\subseteq\mathbb{Q}^{d}$ is _$p$ -discrete with respect to
$\mathbf{w}$_ if the following conditions are satisfied.

  1. (a)

There exists an open cone $\sigma$ containing $\mathbf{w}$ for which the set
$\\{\mathbf{w}^{\prime}\cdot\mathbf{a}:\mathbf{a}\in A\\}$ is well ordered
when $\mathbf{w}^{\prime}\in\sigma\cap\mathbb{R}^{d}_{li}$.

  2. (b)

There is $N>0$, $\boldsymbol{\gamma}\in\mathbb{Q}^{d}$, a pointed cone $C$
with $\mathbf{w}\in\operatorname{int}(C^{\vee})$ such that

$A\subseteq(\boldsymbol{\gamma}+C)\cap\left(\bigcup_{j\geq
0}\frac{1}{Np^{j}}\mathbb{Z}^{d}\right).$

  3. (c)

For any sequence $\\{\mathbf{a}_{i}\\}$ in $A$, if the sequence
$\\{\mathbf{w}\cdot\mathbf{a}_{i}\\}$ converges in $\mathbb{R}$ then
$\\{\mathbf{a}_{i}\\}$ converges to a point of $\mathbb{Q}^{d}$.

  4. (d)

For all $\boldsymbol{\gamma}^{\prime}\in\mathbb{Q}^{d}$ there is an open cone
$\sigma_{\boldsymbol{\gamma}^{\prime}}$ containing $\mathbf{w}$ for which for
all
$\mathbf{w}^{\prime}\in\sigma_{\boldsymbol{\gamma}^{\prime}}\cap\mathbb{R}^{d}_{li}$,
we have
$A^{+}_{\boldsymbol{\gamma}^{\prime},\mathbf{w}^{\prime}}=A^{+}_{\boldsymbol{\gamma}^{\prime},\mathbf{w}}$
and
$A^{-}_{\boldsymbol{\gamma}^{\prime},\mathbf{w}^{\prime}}=A^{-}_{\boldsymbol{\gamma}^{\prime},\mathbf{w}}$.

For a given $p$-discrete set $A$ we may choose the closure of $\sigma$ in
condition a and the dual cone $C^{\vee}$ of $C$ in condition b to coincide, at
the expense of at least one of them not being the largest possible, since
$\sigma$ can be replaced with a smaller open cone and $C$ can be replaced with
a larger pointed cone. In condition d it is possible that a different
$\sigma_{\boldsymbol{\gamma}^{\prime}}$ is needed for each
$\boldsymbol{\gamma}^{\prime}$. Note that condition d is equivalent to the
existence of open cones $\sigma_{\boldsymbol{\gamma}^{\prime}}$ containing
$\mathbf{w}$ such that that
$A^{-}_{\boldsymbol{\gamma}^{\prime},\mathbf{w}}\subseteq\boldsymbol{\gamma}^{\prime}-\sigma_{\boldsymbol{\gamma}^{\prime}}^{\vee}$,
and
$A^{+}_{\boldsymbol{\gamma}^{\prime},\mathbf{w}}\subseteq\boldsymbol{\gamma}^{\prime}+\sigma_{\boldsymbol{\gamma}^{\prime}}^{\vee}$.

######  Definition 2.3.

For fixed $\mathbf{w}\in\mathbb{R}^{d}_{li}$, we define
$\mathcal{A}_{\mathbf{w}}$ to be the set

($*$) $\mathcal{A}_{\mathbf{w}}=\\{A\subset\mathbb{Q}^{d}:A\text{ is
}p\text{-discrete with respect to }\mathbf{w}\\}.$

######  Remark 2.4.

Saavedra’s field family [Saavedra]*Proposition 5.1 satisfies condition b, and
also a weaker version of condition a, where the well ordering is required only
for $\mathbf{w}$. The stronger condition a is needed in Section 4 to show that
some specialization maps $t_{i}=x^{n_{i}}$ are well defined. Condition d is
needed in the proof of algebraic closure in Section 3 and is also used in
Section 4. Condition c is needed in the proof of Lemma 2.6 to show that these
series form a field family.

In the rest of this section we show that $\mathcal{A}_{\mathbf{w}}$ is a field
family with respect to $(\mathbb{Q}^{d},\preceq_{\mathbf{w}})$.

######  Lemma 2.5.

Fix $\mathbf{w}\in\mathbb{R}^{d}_{li}$, and
$\boldsymbol{\gamma}\in\mathbb{Q}^{d}$. Suppose $A\in\mathcal{A}_{\mathbf{w}}$
with $\mathbf{w}\cdot\mathbf{a}\geq 0$ for all $\mathbf{a}\in A$. Let $S(A)$
be the semigroup generated by $A$ under addition in $\mathbb{Q}^{d}$ and let
$S(A)^{-}_{\boldsymbol{\gamma},\mathbf{w}}$ be as in Definition 2.2. Then
there is $M>0$ such that for any nonzero
$\mathbf{s}=\sum_{i=1}^{m}\mathbf{a}_{i}\in
S(A)^{-}_{\boldsymbol{\gamma},\mathbf{w}}$, where $\mathbf{a}_{i}\in
A\setminus\\{0\\}$, the number $m$ of summands is at most $M$.

###### Proof.

We have
$\mathbf{w}\cdot\mathbf{s}=\sum_{i=1}^{m}\mathbf{w}\cdot\mathbf{a}_{i}$, so

$0<\min_{i}\\{\mathbf{w}\cdot\mathbf{a}_{i}\\}\leq\frac{\mathbf{w}\cdot\mathbf{s}}{m}<\frac{\mathbf{w}\cdot\boldsymbol{\gamma}}{m},$

where the last inequality holds because $\mathbf{s}\in
S(A)^{-}_{\boldsymbol{\gamma},\mathbf{w}}$. If for all $m>0$ there is
$\mathbf{s}\in S(A)^{-}_{\boldsymbol{\gamma},\mathbf{w}}$ that is the sum of
at least $m$ nonzero terms, we would get a contradiction to well-ordering of
the set $\\{\mathbf{w}\cdot\mathbf{a}\colon\mathbf{a}\in A\\}$ given in
condition a. ∎

######  Lemma 2.6.

Let $A\in\mathcal{A}_{\mathbf{w}}$ and let $S(A)$ be the semigroup generated
by elements of $A$ under addition. Suppose that $\mathbf{w}\cdot\mathbf{a}\geq
0$ for all $\mathbf{a}\in A$, and conditions a, b, c of $p$-discreteness hold
for $S(A)$. Then $S(A)$ also satisfies condition d, so $S(A)$ is in
$\mathcal{A}_{\mathbf{w}}$.

###### Proof.

Fix $\boldsymbol{\gamma}^{\prime}\in\mathbb{Q}^{d}$. It suffices to show there
are open cones $\sigma_{1}$ and $\sigma_{2}$ containing $\mathbf{w}$ for which
$S(A)^{+}_{\boldsymbol{\gamma}^{\prime},\mathbf{w}}\subseteq
S(A)^{+}_{\boldsymbol{\gamma}^{\prime},\mathbf{w}^{\prime}}$ for all
$\mathbf{w}^{\prime}\in\sigma_{1}\cap\mathbb{R}^{d}_{li}$ and
$S(A)^{-}_{\boldsymbol{\gamma}^{\prime},\mathbf{w}}\subseteq
S(A)^{-}_{\boldsymbol{\gamma}^{\prime},\mathbf{w}^{\prime}}$ for all
$\mathbf{w}^{\prime}\in\sigma_{2}\cap\mathbb{R}^{d}_{li}$.

We first show the existence of $\sigma_{1}$. Since
$\\{\mathbf{w}\cdot\mathbf{a}:\mathbf{a}\in
S(A)^{+}_{\boldsymbol{\gamma}^{\prime},\mathbf{w}}\\}$ is well ordered by a
there is $\mathbf{a}\in S(A)^{+}_{\boldsymbol{\gamma}^{\prime},\mathbf{w}}$
achieving the minimum value of $\mathbf{w}\cdot\mathbf{a}$, so

$\delta:=\min\\{\mathbf{w}\cdot\mathbf{a}:\mathbf{a}\in
S(A)^{+}_{\boldsymbol{\gamma}^{\prime},\mathbf{w}}\\}-\mathbf{w}\cdot\boldsymbol{\gamma}^{\prime}$

is positive. By condition b, we have $S(A)\subseteq\boldsymbol{\gamma}+C$ for
a pointed cone $C$ with $\mathbf{w}\in\operatorname{int}(C^{\vee})$ and
$\boldsymbol{\gamma}\in\mathbb{Q}^{d}$. If
$\mathbf{w}\cdot\boldsymbol{\gamma}^{\prime}\leq\mathbf{w}\cdot\boldsymbol{\gamma}$
then we may take $\sigma_{1}$ to be the interior of
$\\{\mathbf{x}:\mathbf{x}\cdot\boldsymbol{\gamma}\geq\mathbf{w}^{\prime}\cdot\boldsymbol{\gamma}^{\prime}\\}\cap
C^{\vee}$. We now suppose that
$\mathbf{w}\cdot\boldsymbol{\gamma}^{\prime}>\mathbf{w}\cdot\boldsymbol{\gamma}$.
We may assume that $C$ is full-dimensional here, so there is a rational point
$\tilde{\boldsymbol{\gamma}}$ in
$\boldsymbol{\gamma}+C\cap\\{\mathbf{x}\in\mathbb{R}^{d}\colon\mathbf{w}\cdot\boldsymbol{\gamma}^{\prime}<\mathbf{w}\cdot\mathbf{x}<\mathbf{w}\cdot\boldsymbol{\gamma}^{\prime}+\delta\\}$.
Let $C^{\prime}$ be the closure of the cone generated by
$\\{\mathbf{a}-\tilde{\boldsymbol{\gamma}}:\mathbf{a}\in
S(A)^{+}_{\boldsymbol{\gamma}^{\prime},\mathbf{w}}\\}$. The cone $C^{\prime}$
is pointed, as it is contained in the cone with vertex
$\tilde{\boldsymbol{\gamma}}$ over the intersection
$\boldsymbol{\gamma}+C\cap\\{\mathbf{x}:\mathbf{w}\cdot\mathbf{x}=\mathbf{w}\cdot\boldsymbol{\gamma}^{\prime}+\delta\\}$,
which is bounded. It also contains $\mathbf{w}$ in its dual. Thus any open
cone $\sigma_{1}$ containing $\mathbf{w}$ and contained in
$\operatorname{int}({C^{\prime}}^{\vee})\cap\\{\mathbf{x}\in\mathbb{R}^{d}\colon\mathbf{x}\cdot(\tilde{\boldsymbol{\gamma}}-\boldsymbol{\gamma}^{\prime})\geq
0\\}$ will have the property that
$S(A)^{+}_{\boldsymbol{\gamma}^{\prime},\mathbf{w}}\subseteq
S(A)^{+}_{\boldsymbol{\gamma}^{\prime},\mathbf{w}^{\prime}}$ for all
$\mathbf{w}^{\prime}\in\sigma_{1}$.

We next show the existence of $\sigma_{2}$.

By Lemma 2.5 there is a bound $M$ on the number of summands $m$ in any element
$\mathbf{a}\in S(A)^{-}_{\boldsymbol{\gamma}^{\prime},\mathbf{w}}$. Let

$B_{m}=\left\\{(\mathbf{a}_{1},\dots,\mathbf{a}_{m})\in
A^{m}:\sum_{i=1}^{m}\mathbf{a}_{i}\in
S(A)^{-}_{\boldsymbol{\gamma}^{\prime},\mathbf{w}}\right\\}.$

We show by induction on $m$ that there is a cone $C_{m}$ with
$\mathbf{w}\in\operatorname{int}(C_{m}^{\vee})$ and
$\\{\sum_{i=1}^{m}\mathbf{a}_{i}\in
S(A)^{-}_{\boldsymbol{\gamma}^{\prime},\mathbf{w}}\\}\subseteq\boldsymbol{\gamma}^{\prime}-C_{m}$.
Any $\sigma_{2}\subseteq\operatorname{int}(\bigcap_{m=1}^{M}C_{m}^{\vee})$
containing $\mathbf{w}$ will then have the property that
$S(A)^{-}_{\boldsymbol{\gamma}^{\prime},\mathbf{w}}\subseteq
S(A)^{-}_{\boldsymbol{\gamma}^{\prime},\mathbf{w}^{\prime}}$ for all
$\mathbf{w}^{\prime}\in\sigma_{2}$.

The base case is $m=1$, where the claim follows since $B_{1}\subseteq A$, so
$B_{1}\in\mathcal{A}_{\mathbf{w}}$, for which axiom d holds. Now suppose that
$m>1$, and the claim is true for smaller $m$. If

$s:=\sup\left\\{\mathbf{w}\cdot\sum_{i=1}^{m}\mathbf{a}_{i}:(\mathbf{a}_{1},\ldots,\mathbf{a}_{m})\in
B_{m}\right\\}$

is less than $\mathbf{w}\cdot\boldsymbol{\gamma}^{\prime}$, then the closure
$C_{m}$ of the cone generated by
$\\{\boldsymbol{\gamma}^{\prime}-\sum_{i=1}^{m}\mathbf{a}_{i}:(\mathbf{a}_{1},\ldots,\mathbf{a}_{m})\in
B_{m}\\}$ is pointed and has the required form. We may thus assume that
$s=\mathbf{w}\cdot\boldsymbol{\gamma}^{\prime}$.

For each $n>0$, the set

$\mathcal{S}^{(0)}_{n}=\left\\{\mathbf{a}_{1}:(\mathbf{a}_{1},\ldots,\mathbf{a}_{m})\in
B_{m},\mathbf{w}\cdot\sum_{i=1}^{m}\mathbf{a}_{i}>\mathbf{w}\cdot\boldsymbol{\gamma}^{\prime}-1/n\right\\}$

is a subset of $A$, so by axiom a there is some
$\mathbf{s}_{n}^{(0)}\in\mathcal{S}_{n}^{(0)}$ with
$\mathbf{w}\cdot\mathbf{s}_{n}^{(0)}$ minimal. We claim that the set
$\\{\mathbf{w}\cdot\mathbf{s}_{n}^{(0)}:n\geq 1\\}$ is weakly increasing, and
bounded above by $\mathbf{w}\cdot\boldsymbol{\gamma}^{\prime}$. The bound
comes from the fact that $\mathbf{w}\cdot\mathbf{a}_{i}\geq 0$ for all
$\mathbf{a}_{i}$ by assumption, which implies that
$\mathbf{w}\cdot\mathbf{s}_{n}^{(0)}=\mathbf{w}\cdot\mathbf{a}_{1}\leq\mathbf{w}\cdot\sum_{i=1}^{m}\mathbf{a}_{i}<\mathbf{w}\cdot\boldsymbol{\gamma}^{\prime}$.
Thus the sequence $\\{\mathbf{w}\cdot\mathbf{s}_{n}^{(0)}\\}$ converges to
$\ell_{0}\in\mathbb{R}$. We now iterate, at each stage constructing sets

$\mathcal{T}_{j}=\\{(\mathbf{a}_{1},\ldots,\mathbf{a}_{m})\in
B_{m}:\mathbf{w}\cdot\mathbf{a}_{1}>\ell_{j-1}\\},$

and

$\mathcal{S}_{n}^{(j)}=\left\\{\mathbf{a}_{1}:(\mathbf{a}_{1},\ldots,\mathbf{a}_{m})\in\mathcal{T}_{j},\mathbf{w}\cdot\sum_{i=1}^{m}\mathbf{a}_{i}>\mathbf{w}\cdot\boldsymbol{\gamma}^{\prime}-1/n\right\\}.$

with sequences $\\{\mathbf{s}_{n}^{(j)}\\}$, for which
$\mathbf{w}\cdot\mathbf{s}_{n}^{(j)}$ converges to $\ell_{j}$. Note that
$\ell_{j}>\ell_{j-1}$ by construction.

We claim that this process must terminate for some $j$, with
$\sup\\{\mathbf{w}\cdot\sum_{i=1}^{m}\mathbf{a}_{i}:(\mathbf{a}_{1},\ldots,\mathbf{a}_{m})\in\mathcal{T}_{j}\\}<\mathbf{w}\cdot\boldsymbol{\gamma}^{\prime}$.
To see this, suppose that we can construct an infinite sequence

$\ell_{0}<\ell_{1}<\ell_{2}<\ell_{3}<\dots$

For each $j>1$, fix
$0<\epsilon_{j}<\min\\{(\ell_{j+1}-\ell_{j})/2,(\ell_{j}-\ell_{j-1})/2\\}$.
Since for each $j$, the sequence $\\{\mathbf{w}\cdot\mathbf{s}_{n}^{(j)}\\}$
converges to $\ell_{j}$ there is $N_{j}>0$ for which
$|\mathbf{w}\cdot\mathbf{s}_{n}^{(j)}-\ell_{j}|<\epsilon_{j}$ for $n>N_{j}$.
Fix $n_{j}>N_{j}$ with $1/n_{j}<\epsilon_{j}$ and pick
$\mathbf{c}_{j}=(\mathbf{a}_{1},\ldots,\mathbf{a}_{m})\in\mathcal{T}_{j}$ with
$\mathbf{a}_{1}=\mathbf{s}_{n_{j}}^{(j)}$. Then
$\mathbf{w}\cdot\sum_{i=1}^{m}\mathbf{a}_{i}>\mathbf{w}\cdot\boldsymbol{\gamma}^{\prime}-\epsilon_{j}$.
Because $\sum_{i=1}^{m}\mathbf{a}_{i}\in
S(A)^{-}_{\boldsymbol{\gamma}^{\prime},\mathbf{w}}$, we also have the
condition that
$\mathbf{w}\cdot\sum_{i=1}^{m}\mathbf{a}_{i}<\mathbf{w}\cdot\boldsymbol{\gamma}^{\prime}$.
Subtracting
$\mathbf{w}\cdot\mathbf{s}_{n_{j}}^{(j)}=\mathbf{w}\cdot\mathbf{a}_{1}$ we get

$\mathbf{w}\cdot\boldsymbol{\gamma}^{\prime}-\epsilon_{j}-\mathbf{w}\cdot\mathbf{s}_{n_{j}}^{(j)}<\sum_{i=2}^{m}\mathbf{w}\cdot\mathbf{a}_{i}<\mathbf{w}\cdot\boldsymbol{\gamma}^{\prime}-\mathbf{w}\cdot\mathbf{s}_{n_{j}}^{(j)}.$

Hence, since
$-\ell_{j}+\epsilon_{j}>-\mathbf{w}\cdot\mathbf{s}_{n_{j}}^{(j)}\geq-\ell_{j}$
and by the choice of $\epsilon_{j}$, we have

$\mathbf{w}\cdot\mathbf{\boldsymbol{\gamma}}^{\prime}-\ell_{j+1}<\mathbf{w}\cdot\mathbf{\boldsymbol{\gamma}}^{\prime}-\ell_{j}-\epsilon_{j}<\sum_{i=2}^{m}\mathbf{w}\cdot\mathbf{a}_{i}<\mathbf{w}\cdot\boldsymbol{\gamma}^{\prime}-\ell_{j}+\epsilon_{j}<\mathbf{w}\cdot\boldsymbol{\gamma}^{\prime}-\ell_{j-1}.$

But this implies that the subset
$\\{\sum_{i=2}^{m}\mathbf{a}_{i}:\mathbf{c}_{j}=(\mathbf{a}_{1},\ldots,\mathbf{a}_{m})\\}$
of $S(A)$ is not well ordered, contradicting our assumption that condition a
holds. From this contradiction we conclude that the process terminates, so
there is $j$ for which
$\sup\\{\mathbf{w}\cdot\sum_{i=1}^{m}\mathbf{a}_{i}:(\mathbf{a}_{1},\ldots,\mathbf{a}_{m})\in\mathcal{T}_{j}\\}<\mathbf{w}\cdot\boldsymbol{\gamma}^{\prime}$.
Write $L$ for this $j$ and set

$\epsilon:=\mathbf{w}\cdot\boldsymbol{\gamma}^{\prime}-\sup\left\\{\mathbf{w}\cdot\sum_{i=1}^{m}\mathbf{a}_{i}:(\mathbf{a}_{1},\ldots,\mathbf{a}_{m})\in\mathcal{T}_{L}\right\\},$

which is positive.

For each $0\leq j\leq L-1$, set

$\mathcal{T}_{j}^{\prime}=\left\\{(\mathbf{a}_{1},\ldots,\mathbf{a}_{m})\in
B_{m}:\mathbf{w}\cdot\sum_{i=2}^{m}\mathbf{a}_{i}>\mathbf{w}\cdot\boldsymbol{\gamma}^{\prime}-\ell_{j}\right\\}.$

The set
$\\{\mathbf{w}\cdot\sum_{i=2}^{m}\mathbf{a}_{i}:(\mathbf{a}_{1},\ldots,\mathbf{a}_{m})\in\mathcal{T}_{j}^{\prime}\\}$
is well ordered by a, so

$\mu_{j}:=\min\left\\{\mathbf{w}~{}\cdot~{}\sum_{i=2}^{m}\mathbf{a}_{i}-\mathbf{w}\cdot\boldsymbol{\gamma}^{\prime}+\ell_{j}:(\mathbf{a}_{1},\ldots,\mathbf{a}_{m})\in\mathcal{T}^{\prime}_{j}\right\\}$

exists and is positive. Choose $n\in\mathbb{N}$ with
$1/n<\min_{j}(\mu_{j},\epsilon)$, and $n^{\prime}>n$ with
$\ell_{j}-\mathbf{w}\cdot\mathbf{s}_{n^{\prime}}^{(j)}<1/n$ for all $0\leq
j<L$.

Set $\mathcal{T}_{0}=B_{m}$, and fix $0\leq j<L$. Since $\ell_{j}$ is the
limit of the sequence $\\{\mathbf{w}\cdot\mathbf{s}_{n}^{(j)}\\}$, for each
$j$, by condition c we have
$\ell_{j}=\mathbf{w}\cdot\tilde{\boldsymbol{\gamma}}_{j}$ for
$\tilde{\boldsymbol{\gamma}}_{j}\in\mathbb{Q}^{d}$. We first consider the case
that
$(\mathbf{a}_{1},\dots,\mathbf{a}_{m})\in\mathcal{T}_{j}\setminus\mathcal{T}_{j+1}$
satisfies

(2)
$\mathbf{w}\cdot\sum_{i=1}^{m}\mathbf{a}_{i}>\mathbf{w}\cdot\boldsymbol{\gamma}^{\prime}-1/n^{\prime}.$

In that case
$\mathbf{w}\cdot\mathbf{a}_{1}\geq\mathbf{w}\cdot\mathbf{s}_{n^{\prime}}^{(j)}>\ell_{j}-1/n$.
Thus
$\mathbf{w}\cdot\sum_{i=2}^{m}\mathbf{a}_{i}<\mathbf{w}\cdot\boldsymbol{\gamma}^{\prime}-\ell_{j}+1/n<\mathbf{w}\cdot\boldsymbol{\gamma}^{\prime}-\ell_{j}+\mu_{j}$,
so by the definition of $\mu_{j}$ we have
$\mathbf{w}\cdot\sum_{i=2}^{m}\mathbf{a}_{i}\leq\mathbf{w}\cdot\boldsymbol{\gamma}^{\prime}-\ell_{j}$,
and thus
$\mathbf{w}\cdot\sum_{i=2}^{m}\mathbf{a}_{i}\leq\mathbf{w}\cdot\boldsymbol{\gamma}^{\prime}-\mathbf{w}\cdot\tilde{\boldsymbol{\gamma}}_{j}$.

By induction there is a cone $C_{j,m-1}$ with
$\mathbf{w}\in\operatorname{int}(C_{j,m-1}^{\vee})$ and

$\left\\{\sum_{i=2}^{m}\mathbf{a}_{i}:(\mathbf{a}_{1},\ldots,\mathbf{a}_{m})\in
B_{m},\mathbf{w}\cdot\sum_{i=2}^{m}\mathbf{a}_{i}\leq\mathbf{w}\cdot\boldsymbol{\gamma}^{\prime}-\mathbf{w}\cdot\tilde{\boldsymbol{\gamma}}_{j}\right\\}\subseteq\boldsymbol{\gamma}^{\prime}-\tilde{\boldsymbol{\gamma}}_{j}-C_{j,m-1}.$

By d for $A$ there is an open cone $C^{\prime}_{j}$ with
$\mathbf{w}\in\operatorname{int}({C^{\prime}_{j}}^{\vee})$ and
$\\{\mathbf{a}_{1}:\mathbf{w}\cdot\mathbf{a}_{1}\leq\ell_{j}=\mathbf{w}\cdot\tilde{\boldsymbol{\gamma}}_{j}\\}\subseteq\tilde{\boldsymbol{\gamma}}_{j}-C^{\prime}_{j}$.
Thus if
$(\mathbf{a}_{1},\dots,\mathbf{a}_{m})\in\mathcal{T}_{j}\setminus\mathcal{T}_{j+1}$
with
$\mathbf{w}\cdot\sum_{i=1}^{m}\mathbf{a}_{i}>\mathbf{w}\cdot\boldsymbol{\gamma}^{\prime}-1/n^{\prime}$
then

$\sum_{i=1}^{m}\mathbf{a}_{i}\in\boldsymbol{\gamma}^{\prime}-(C_{j}^{\prime}+C_{j,m-1}).$

Let $C^{\prime\prime}$ be the Minkowski sum
$\sum_{j=0}^{L-1}(C_{j}^{\prime}+C_{j,m-1})$. Note that for
$(\mathbf{a}_{1},\dots,\mathbf{a}_{m})\in\mathcal{T}_{L}$ we have
$\mathbf{w}\cdot\sum_{i=1}^{m}\mathbf{a}_{i}<\mathbf{w}\cdot\boldsymbol{\gamma}^{\prime}-1/n^{\prime}$.
Thus for $(\mathbf{a}_{1},\dots,\mathbf{a}_{m})\in B_{m}$ with
$\mathbf{w}\cdot\sum_{i=1}^{m}\mathbf{a}_{i}>\mathbf{w}\cdot\boldsymbol{\gamma}^{\prime}-1/n^{\prime}$
we have

$\sum_{i=1}^{m}\mathbf{a}_{i}\in\boldsymbol{\gamma}^{\prime}-C^{\prime\prime}.$

The closure of the cone $C^{\prime\prime\prime}$ generated by

$\left\\{\boldsymbol{\gamma}^{\prime}-\sum_{i=1}^{m}\mathbf{a}_{i}:(\mathbf{a}_{1},\ldots,\mathbf{a}_{m})\in
B_{m},\mathbf{w}\cdot\sum_{i=1}^{m}\mathbf{a}_{i}\leq\mathbf{w}\cdot\boldsymbol{\gamma}^{\prime}-1/n^{\prime}\right\\}$

is pointed,
$\mathbf{w}\in\operatorname{int}({C^{\prime\prime\prime}}^{\vee})$, and
$\\{\sum_{i=1}^{m}\mathbf{a}_{i}:(\mathbf{a}_{1},\ldots,\mathbf{a}_{m})\in
B_{m},\mathbf{w}\cdot\sum_{i=1}^{m}\mathbf{a}_{i}\leq\mathbf{w}\cdot\boldsymbol{\gamma}^{\prime}-1/n^{\prime}\\}\subseteq\boldsymbol{\gamma}^{\prime}-C^{\prime\prime\prime}$.

Finally, let $C_{m}=C^{\prime\prime}+C^{\prime\prime\prime}$. Then
$\mathbf{w}\in\operatorname{int}(C_{m}^{\vee})$, and for all
$(\mathbf{a}_{1},\ldots,\mathbf{a}_{m})\in B_{m}$ with
$\mathbf{w}\cdot\sum_{i=1}^{m}\mathbf{a}_{i}<\mathbf{w}\cdot\boldsymbol{\gamma}^{\prime}$,
we have $\sum_{i=1}^{m}\mathbf{a}_{i}\in\boldsymbol{\gamma}^{\prime}-C_{m}$ as
required. ∎

######  Theorem 2.7.

The set $\mathcal{A}_{\mathbf{w}}$ is a field family with respect to
$(\mathbb{Q}^{d},\preceq_{\mathbf{w}})$.

###### Proof.

We check each of the six axioms of a field family given in Definition 2.1.

_Axiom (i)_ : Each $A\in\mathcal{A}_{\mathbf{w}}$ is well ordered by condition
a of Definition 2.2.

_Axiom (ii)_ : Every $\boldsymbol{\gamma}\in\mathbb{Q}^{d}$ is in
$\mathcal{A}_{\mathbf{w}}$.

_Axiom (iii)_ : Fix $A,B\in\mathcal{A}_{\mathbf{w}}$. Condition b of
$p$-discreteness holds for $A\cup B$ by [Saavedra]*Proposition 5.1. Let
$\sigma_{A},\sigma_{B}$ be the respective open cones containing $\mathbf{w}$
guaranteed by condition a of $p$-discreteness for $A$ and $B$. The open cone
$\sigma=\sigma_{A}\cap\sigma_{B}$ makes condition a hold for $A\cup B$.
Similarly, for condition d we may use the intersection of the respective
guaranteed open cones for $A$ and $B$.

For c, suppose $\\{\mathbf{c}_{i}\\}$ is a sequence in $A\cup B$ with
$\\{\mathbf{w}\cdot\mathbf{c}_{i}\\}$ converging to $L\in\mathbb{R}$. Let
$\\{\mathbf{a}_{j}\\}$ and $\\{\mathbf{b}_{j}\\}$ be the subsequences
consisting of points in $A$ and in $B$, respectively. If one of these
subsequences is finite, then $\\{\mathbf{c}_{i}\\}$ converges to the limit of
the other sequence. Otherwise, $\\{\mathbf{w}\cdot\mathbf{a}_{j}\\}$ and
$\\{\mathbf{w}\cdot\mathbf{b}_{j}\\}$ also converge to $L$, so
$\\{\mathbf{a}_{j}\\}$ converges to $\mathbf{a}$ and $\\{\mathbf{b}_{j}\\}$
converges to $\mathbf{b}$ with $\mathbf{a},\mathbf{b}\in\mathbb{Q}^{d}$. We
have $\mathbf{w}\cdot\mathbf{a}=\mathbf{w}\cdot\mathbf{b}=L$, so
$\mathbf{w}\cdot(\mathbf{a}-\mathbf{b})=0$. Since
$\mathbf{w}\in\mathbb{R}^{d}_{li}$, this implies that
$\mathbf{a}-\mathbf{b}=0$, so $\mathbf{a}=\mathbf{b}$, and
$\\{\mathbf{c}_{i}\\}$ converges to the point $\mathbf{a}=\mathbf{b}$ in
$\mathbb{Q}^{d}$ as required.

_Axiom (iv)_ : All conditions of $p$-discreteness are inherited from $A$ by
$B\subseteq A$.

_Axiom (v)_ : All conditions for a set $A\subseteq\mathbb{Q}^{d}$ to be
$p$-discrete are invariant under translation by a point in $\mathbb{Q}^{d}$.

_Axiom (vi)_ : Fix $A\in\mathcal{A}_{\mathbf{w}}$ with
$\mathbf{w}\cdot\mathbf{a}\geq 0$ for all $\mathbf{a}\in A$. Let $S(A)$ be the
semigroup generated by $A$ under addition. Let $\sigma$ be as in condition a
of $p$-discreteness for $A$, and let $\sigma_{\mathbf{0}}$ be as in condition
d for $\mathbf{0}$. For any
$\mathbf{w}^{\prime}\in\sigma\cap\sigma_{\mathbf{0}}$ we have
$A^{+}_{0,\mathbf{w}^{\prime}}=A^{+}_{0,\mathbf{w}}=A$, since
$\mathbf{w}\cdot\mathbf{a}\geq 0$ for all $\mathbf{a}\in A$, and $A$ is
$p$-discrete. Thus the fact that $S(A)$ is well ordered with respect to
$\mathbf{w}^{\prime}$ follows from [Neumann]*Theorem 3.4. Condition b for
$S(A)$ follows as in [Saavedra]*Proposition 5.1.

We now prove condition c holds. Let $\\{\mathbf{s}_{i}\\}$ be a sequence in
$S(A)$ such that $\\{\mathbf{w}\cdot\mathbf{s}_{i}\\}$ converges to some
$L\in\mathbb{R}$. We want to show that $\\{\mathbf{s}_{i}\\}$ converges to
some $\mathbf{s}\in\mathbb{Q}^{d}$. Pick
$\boldsymbol{\gamma}\in\mathbb{Q}^{d}$ such that
$\mathbf{w}\cdot\boldsymbol{\gamma}>L$. There exists an integer $N$ such that
for all $i>N$, we have
$|\mathbf{w}\cdot\mathbf{s}_{i}-L|<\mathbf{w}\cdot\boldsymbol{\gamma}-L$.
Consequently, for all $i>N$ we have $\mathbf{s}_{i}\in
S(A)^{-}_{\boldsymbol{\gamma},\mathbf{w}}$. When
$\mathbf{s}_{i}\neq\mathbf{0}$, Lemma 2.5 says that the number $m$ of summands
of $\mathbf{s}_{i}=\sum_{j=1}^{m}\mathbf{a}^{(j)}_{i}$ is bounded by some
$M>0$. When $\mathbf{s}_{i}=\mathbf{0}$, the assumption that
$\mathbf{w}\cdot\mathbf{a}\geq 0$ for all $\mathbf{a}\in A$ implies that
$\mathbf{s}_{i}=\sum_{j=1}^{m}\mathbf{a}^{(j)}_{i}$ has just one summand,
namely $\mathbf{0}\in A$. Thus by passing to a subsequence
$\\{\mathbf{s}_{n(i)}\\}$ of $\\{\mathbf{s}_{i}\\}$ we may assume that each
$\mathbf{s}_{n(i)}$ is a sum of exactly $m$ terms
$\mathbf{s}_{n(i)}=\sum_{j=1}^{m}\mathbf{a}_{i}^{(j)}$ for some $m\leq M$, and
the sequence $\\{\mathbf{w}\cdot\mathbf{a}_{i}^{(j)}\\}_{i}$ is weakly
increasing for each $1\leq j\leq m$. By hypothesis we have
$0\leq\mathbf{w}\cdot\mathbf{a}_{i}^{(j)}<\mathbf{w}\cdot\boldsymbol{\gamma}$.
Hence for each fixed $1\leq j\leq m$ the weakly increasing and bounded
sequence $\\{\mathbf{w}\cdot\mathbf{a}_{i}^{(j)}\\}_{i}$ converges. Since $A$
is $p$-discrete, each $\\{\mathbf{a}_{i}^{(j)}\\}_{i}$ converges to some
$\mathbf{a}^{(j)}\in\mathbb{Q}^{d}$, so $\\{\mathbf{s}_{n(i)}\\}$ converges to
$\mathbf{s}=\sum_{j=1}^{m}\mathbf{a}^{(j)}\in\mathbb{Q}^{d}$.

So far we have shown that $\\{\mathbf{s}_{i}\\}$ has a subsequence converging
to $\mathbf{s}\in\mathbb{Q}^{d}$. We claim that the entire sequence
$\\{\mathbf{s}_{i}\\}$ also converges to $\mathbf{s}$. Suppose there exists an
$\epsilon>0$ for which for all $n\in\mathbb{N}$, there is an index $i_{n}>n$
such that $|\mathbf{s}_{i_{n}}-\mathbf{s}|>\epsilon$. Note that
$\\{\mathbf{w}\cdot\mathbf{s}_{i_{n}}\\}_{n}$ also converges to $L$. Hence, by
applying the same argument as above, we deduce that the sequence
$\\{\mathbf{s}_{i_{n}}\\}_{n}$ itself has a convergent subsequence to
$\mathbf{s}$, contradicting the $\epsilon$ distance of the
$\mathbf{s}_{i_{n}}$ away from $\mathbf{s}$.

Condition d of $p$-discreteness now holds by Lemma 2.6, which completes the
proof. ∎

######  Definition 2.8.

Let $\mathbbm{k}$ be an algebraically closed field of characteristic $p>0$ and
let $\mathbf{w}\in\mathbb{R}^{d}_{li}$. We denote by $K_{\mathbf{w}}$ the
field $\mathbbm{k}^{\mathbb{Q}^{d}}(\mathcal{A}_{\mathbf{w}})$ of $p$-discrete
series in direction $\mathbf{w}$ with coefficients in $\mathbbm{k}$, with
variables $t_{1},\dots,t_{d}$. Specifically, the elements of $K_{\mathbf{w}}$
are formal power series in variables $t_{1},\dots t_{d}$, with exponents in
the field family $\mathcal{A}_{\mathbf{w}}$ and coefficients in $\mathbbm{k}$.

##  3\. The field of p-discrete series is algebraically closed

In this section, we prove that $K_{\mathbf{w}}$ is algebraically closed. We
begin with some preliminary results.

######  Lemma 3.1.

Let $A\in\mathcal{A}_{\bf w}$, and assume that $\mathbf{w}\cdot\mathbf{a}<0$
for all $\mathbf{a}\in A$. Then there is an $M>0$ such that $|\mathbf{a}|\leq
M$ for all $\mathbf{a}\in A$. That is, $A$ is bounded.

###### Proof.

Since $A\in\mathcal{A}_{\bf w}$, the set $A$ is contained in the polyhedron
$\boldsymbol{\gamma}+C$ for some $\boldsymbol{\gamma}\in\mathbb{Q}^{d}$ and
pointed cone $C$ with $\mathbf{w}\in\operatorname{int}(C^{\vee})$. The
polyhedron
$(\boldsymbol{\gamma}+C)\cap\\{\mathbf{x}:\mathbf{w}\cdot\mathbf{x}\leq 0\\}$
is a polytope, as its facet normals span $\mathbb{R}^{d}$, so it is bounded,
and thus $A$ is also bounded. ∎

######  Lemma 3.2.

Let $A\in\mathcal{A}_{\bf w}$, and assume that $\mathbf{w}\cdot\mathbf{a}<0$
for all $\mathbf{a}\in A$. Then $S:=\bigcup_{i=1}^{\infty}p^{-i}A$ is also in
the field family $\mathcal{A}_{\bf w}$.

###### Proof.

We check that $S$ satisfies the conditions of Definition 2.2.

_Condition a:_ We use similar ideas to [Saavedra, Lemma 5.2].

Since condition a holds for $A$, there exists an open cone $\sigma$ containing
$\mathbf{w}$ for which
$Q_{i,\mathbf{w}^{\prime}}:=\\{\mathbf{w}^{\prime}\cdot(p^{-i}\mathbf{a}):p^{-i}\mathbf{a}\in
p^{-i}A\\}$ is well ordered for any $i\in\mathbb{Z}_{\geq 0}$ and ${\bf
w}^{\prime}\in\sigma\cap\mathbb{R}^{d}_{li}$. Since condition d holds for $A$,
there is an open cone $\sigma_{\mathbf{0}}$ such that
$A^{-}_{\mathbf{0},\mathbf{w}}=\\{\mathbf{a}\in
A:\mathbf{w}\cdot\mathbf{a}<0\\}$ is equal to $A^{-}_{\mathbf{0},\mathbf{z}}$
for any $\mathbf{z}\in\sigma_{\mathbf{0}}\cap\mathbb{R}^{d}_{li}$. Since
$A=A^{-}_{\mathbf{0},\mathbf{w}}$ by assumption, we have that
$\mathbf{a}\cdot\mathbf{z}<0$ for any $\mathbf{a}\in A$ and any
$\mathbf{z}\in\sigma_{\mathbf{0}}$. Set $V=\sigma\cap\sigma_{\mathbf{0}}$.

Now, let ${\mathbf{w}^{\prime}}\in V\cap\mathbb{R}^{d}_{li}$ and assume that
there exists an infinite strictly decreasing sequence
$T:=\\{\mathbf{w}^{\prime}\cdot(p^{-k_{j}}\mathbf{a}_{j})\\}_{j}$, where each
${\mathbf{a}_{j}}\in A$, and each $k_{j}>0$. Then there must be infinitely
many distinct integers $k_{j}$ in this sequence, or else some
$Q_{i,\mathbf{w}^{\prime}}$ would not be well ordered. So, there is a strictly
increasing subsequence $\\{k_{n(j)}\\}_{j}$ of the sequence $\\{k_{j}\\}_{j}$.
Consider the associated subsequence of $T$:

(3)
$\mathbf{w}^{\prime}\cdot(p^{-k_{n(0)}}\mathbf{a}_{n(0)})>\mathbf{w}^{\prime}\cdot(p^{-k_{n(1)}}\mathbf{a}_{n(1)})>\cdots>\mathbf{w}^{\prime}\cdot(p^{-k_{n(j)}}\mathbf{a}_{n(j)})>\cdots.$

Since the set $\\{\mathbf{w}^{\prime}\cdot\mathbf{a}_{n(j)}\\}_{j}$ is well
ordered, as $\mathbf{w}^{\prime}\in V\cap\mathbb{R}^{d}_{li}$, it has a
smallest element. In particular, there must exist indices $s<t$ with
$\mathbf{w}^{\prime}\cdot\mathbf{a}_{n(s)}\leq\mathbf{w}^{\prime}\cdot\mathbf{a}_{n(t)}$.
Finally $\mathbf{w}^{\prime}\cdot\mathbf{a}_{n(t)}$ is negative, as
$\mathbf{w}^{\prime}\in\sigma_{\mathbf{0}}\cap\mathbb{R}^{d}_{li}$, and
$k_{n(s)}<k_{n(t)}$, so we have

$\mathbf{w}^{\prime}\cdot(p^{-k_{n(s)}}\mathbf{a}_{n(s)})\leq\mathbf{w}^{\prime}\cdot(p^{-k_{n(s)}}\mathbf{a}_{n(t)})<\mathbf{w}^{\prime}\cdot(p^{-k_{n(t)}}\mathbf{a}_{n(t)}),$

which contradicts (3). Hence there is no infinite strictly decreasing sequence
$T$, and the set $\\{\mathbf{w}^{\prime}\cdot\mathbf{s}:\mathbf{s}\in S\\}$ is
well ordered.

_Condition b:_ Since $A\in\mathcal{A}_{\bf w}$, there is $N>0$,
$\boldsymbol{\gamma}\in\mathbb{Q}^{d}$, and a pointed cone $C$ such that
$\mathbf{w}\in\operatorname{int}(C^{\vee})$ and $A$ is contained in the
intersection

$(\boldsymbol{\gamma}+C)\cap\left(\bigcup_{j\geq
0}\frac{1}{Np^{j}}\mathbb{Z}^{d}\right).$

Since $\mathbf{w}\cdot\mathbf{a}<0$ for $\mathbf{w}\in A$, we must have that
$\mathbf{w}\cdot\boldsymbol{\gamma}<0$. Let $C_{1}$ be the convex hull of $C$
and the ray spanned by $-\boldsymbol{\gamma}$. This is a pointed cone, and
$\mathbf{w}\in C_{1}^{\vee}$ because $\mathbf{w}\in C^{\vee}$ and
$-\mathbf{w}\cdot\boldsymbol{\gamma}>0$. For $\mathbf{a}\in A$, since
$\mathbf{a}-\boldsymbol{\gamma}\in C$, we have that
$p^{-i}\mathbf{a}-p^{-i}\boldsymbol{\gamma}\in C$, and that
$(p^{-i}\mathbf{a}-p^{-i}\boldsymbol{\gamma})+(1-p^{-i})(-\boldsymbol{\gamma})=p^{-i}\mathbf{a}-\boldsymbol{\gamma}\in
C_{1}.$ Thus, $p^{-i}\mathbf{a}\in\boldsymbol{\gamma}+C_{1}$, and we conclude
that $p^{-i}A$ is contained in

(4) $(\boldsymbol{\gamma}+C_{1})\cap\left(\bigcup_{j\geq
0}\frac{1}{Np^{j}}\mathbb{Z}^{d}\right).$

Since $C_{1}$ does not depend on the particular $i>0$, we conclude that $S$ is
contained in the set (4).

_Condition c:_ Suppose that $\\{p^{-k_{j}}\mathbf{a}_{j}\\}_{j}$ is a sequence
in $S$ with each $\mathbf{a}_{j}\in A$ such that
$\\{\mathbf{w}\cdot(p^{-k_{j}}\mathbf{a}_{j})\\}_{j}$ converges to an element
of $\mathbb{R}$.

If $\\{p^{-k_{j}}\mathbf{a}_{j}\\}_{j}$ is contained in $p^{-i}A$ for some
fixed $i$ then we may apply condition c for $A$ to conclude that the sequence
converges to an element of $\mathbb{Q}^{d}$. If the sequence
$\\{p^{-k_{j}}\mathbf{a}_{j}\\}_{j}$ is contained in a finite union of the
sets $p^{-i}A$, then we may apply the condition c part of the argument in item
(iii) of the proof of Theorem 2.7 to conclude that our original sequence
$\\{p^{-k_{j}}\mathbf{a}_{j}\\}_{j}$ converges to an element of
$\mathbb{Q}^{d}$.

Assume that the sequence is not contained in a finite union of sets $p^{-i}A$.
Then, there is a strictly increasing subsequence $\\{k_{n(j)}\\}_{j}$ of the
sequence $\\{k_{j}\\}_{j}$. Since the elements of $\\{\mathbf{a}_{j}\\}_{j}$
are bounded in length by Lemma 3.1, the associated subsequence
$\\{p^{-k_{n(j)}}\mathbf{a}_{n(j)}\\}_{j}$ of
$\\{p^{-k_{j}}\mathbf{a}_{j}\\}_{j}$ converges to $\mathbf{0}$. Therefore,
since $\\{\mathbf{w}\cdot(p^{-k_{j}}\mathbf{a}_{j})\\}_{j}$ converges by
assumption, it must converge to $0$. We will show that this forces our
original sequence $\\{p^{-k_{j}}\mathbf{a}_{j}\\}_{j}$ to converge to
$\mathbf{0}$.

Suppose otherwise. Then there is an $\epsilon>0$ and a subsequence
$\\{p^{-k_{m(j)}}\mathbf{a}_{m(j)}\\}_{j}$ of
$\\{p^{-k_{j}}\mathbf{a}_{j}\\}_{j}$ such that
$p^{-k_{m(j)}}|\mathbf{a}_{m(j)}|>\epsilon$ for all $j\geq 0$. By Lemma 3.1,
there exists an $M>0$ such that $|\mathbf{a}_{m(j)}|<M$ for each $j\geq 0$.
Consequently, $p^{-k_{m(j)}}M>p^{-k_{m(j)}}|\mathbf{a}_{m(j)}|>\epsilon$, and
we conclude that $p^{-k_{m(j)}}>\epsilon/M$ for all $j\geq 0$.

Because $\\{\mathbf{w}\cdot(p^{-k_{m(j)}}\mathbf{a}_{m(j)})\\}_{j}$ converges
to $0$, and the coefficients $p^{-k_{m(j)}}$ are bounded below by
$\epsilon/M$, we have that $\\{\mathbf{w}\cdot\mathbf{a}_{m(j)}\\}$ converges
to $0$. As $A\in\mathcal{A}_{\mathbf{w}}$, it follows that
$\\{\mathbf{a}_{m(j)}\\}_{j}$ converges to an element
$\mathbf{z}\in\mathbb{Q}^{d}$ such that $\mathbf{w}\cdot\mathbf{z}=0$. But the
only $\mathbf{z}\in\mathbb{Q}^{d}$ such that $\mathbf{w}\cdot\mathbf{z}=0$ is
$\mathbf{z}=\mathbf{0}$ since $\mathbf{w}\in\mathbb{R}^{d}_{li}$. This is a
contradiction. Hence our original sequence
$\\{p^{-k_{j}}\mathbf{a}_{j}\\}_{j}$ converges to $\mathbf{0}$.

_Condition d:_ Since $A$ satisfies condition d, for each
$\boldsymbol{\delta}\in\mathbb{Q}^{d}$, there is an open cone
$\sigma_{\boldsymbol{\delta}}$ containing $\mathbf{w}$ such that for all
$\mathbf{w}^{\prime}\in\sigma_{\boldsymbol{\delta}}\cap\mathbb{R}^{d}_{li}$,
we have
$A^{+}_{\boldsymbol{\delta},\mathbf{w}^{\prime}}=A^{+}_{\boldsymbol{\delta},\mathbf{w}}$
and
$A^{-}_{\boldsymbol{\delta},\mathbf{w}^{\prime}}=A^{-}_{\boldsymbol{\delta},\mathbf{w}}$.
Fix $\boldsymbol{\gamma}^{\prime}\in\mathbb{Q}^{d}$. We consider three cases:
(i) $\mathbf{w}\cdot\mathbf{\boldsymbol{\gamma}}^{\prime}>0$, (ii)
$\mathbf{w}\cdot\boldsymbol{\gamma}^{\prime}=0$, and (iii)
$\mathbf{w}\cdot\mathbf{\boldsymbol{\gamma}}^{\prime}<0$.

When $\mathbf{w}\cdot\mathbf{\boldsymbol{\gamma}}^{\prime}>0$, we have
$S=S^{-}_{\boldsymbol{\gamma}^{\prime},\mathbf{w}}$, and
$S^{+}_{\mathbf{w},\mathbf{\boldsymbol{\gamma}}^{\prime}}=\emptyset$, since
$\mathbf{w}\cdot\mathbf{s}<0$ for all $\mathbf{s}\in S$. Let
$V:=\\{\mathbf{v}\in\mathbb{R}^{d}:\mathbf{v}\cdot\boldsymbol{\gamma}^{\prime}>0\\}$.
Then, for $\mathbf{a}\in A$ and
$\mathbf{w}^{\prime}\in({\sigma}_{\mathbf{0}}\cap V)\cap\mathbb{R}^{d}_{li}$,
we have $\mathbf{w}^{\prime}\cdot\mathbf{a}<0$ since
$A=A^{-}_{\mathbf{0},\mathbf{w}}=A^{-}_{\mathbf{0},\mathbf{w}^{\prime}}$. So,
$\mathbf{w}^{\prime}\cdot\mathbf{s}<0$ for any $\mathbf{s}\in S$. Since
$\mathbf{w}^{\prime}\in V$, we have
$\mathbf{w}^{\prime}\cdot\mathbf{s}<\mathbf{w}^{\prime}\cdot\mathbf{\boldsymbol{\gamma}}^{\prime}$
for every $\mathbf{s}\in S$. Hence,
$S=S^{-}_{\boldsymbol{\gamma}^{\prime},\mathbf{w}^{\prime}}$ and
$S^{+}_{\boldsymbol{\gamma}^{\prime},\mathbf{w}^{\prime}}=\emptyset$. We
conclude that for each ${\mathbf{w}^{\prime}}\in(\sigma_{\mathbf{0}}\cap
V)\cap\mathbb{R}^{d}_{li}$, we have
$S^{+}_{\boldsymbol{\gamma}^{\prime},\mathbf{w}^{\prime}}=S^{+}_{\boldsymbol{\gamma}^{\prime},\mathbf{w}}$
and
$S^{-}_{\boldsymbol{\gamma}^{\prime},\mathbf{w}^{\prime}}=S^{-}_{\boldsymbol{\gamma}^{\prime},\mathbf{w}}$.

Next consider the case when $\mathbf{w}\cdot\boldsymbol{\gamma}^{\prime}=0$.
Then $\boldsymbol{\gamma}^{\prime}=\mathbf{0}$ since
$\mathbf{w}\in\mathbb{R}^{d}_{li}$. Then for any
$\mathbf{w}^{\prime}\in\sigma_{\mathbf{0}}\cap\mathbb{R}^{d}_{li}$, we have
$S=S^{-}_{\mathbf{0},\mathbf{w}}=S^{-}_{\mathbf{0},\mathbf{w}^{\prime}}$ and
$\emptyset=S^{+}_{\mathbf{0},\mathbf{w}}=S^{+}_{\mathbf{0},\mathbf{w}^{\prime}}$.

Finally, consider the case $\mathbf{w}\cdot\boldsymbol{\gamma}^{\prime}<0$.
Let $i_{0}>0$ be such that for all $i\geq i_{0}$, we have
$\mathbf{w}\cdot(p^{-i}\mathbf{a})>\mathbf{w}\cdot\mathbf{\boldsymbol{\gamma}}^{\prime}$,
for all $\mathbf{a}\in A$. Such an $i_{0}$ exists because
$\mathbf{w}\cdot\boldsymbol{\gamma}^{\prime}<0$ and the set
$\\{\mathbf{w}\cdot\mathbf{a}:\mathbf{a}\in A\\}$ is well ordered.

Let
$\mathbf{w}^{\prime}\in\left(\bigcap_{i=1}^{i_{0}}\sigma_{p^{i}\boldsymbol{\gamma}^{\prime}}\right)\cap\mathbb{R}^{d}_{li}$.
Then,

  1. (1)

for each $1\leq i\leq i_{0}$, we have
$(p^{-i}A)^{+}_{\boldsymbol{\gamma}^{\prime},\mathbf{w}}=(p^{-i}A)^{+}_{\boldsymbol{\gamma}^{\prime},\mathbf{w}^{\prime}}$
and
$(p^{-i}A)^{-}_{\boldsymbol{\gamma}^{\prime},\mathbf{w}}=(p^{-i}A)^{-}_{\boldsymbol{\gamma}^{\prime},\mathbf{w}^{\prime}}$,
because $\mathbf{w}^{\prime}\in\sigma_{p^{i}\boldsymbol{\gamma}^{\prime}}$;

  2. (2)

by the choice of $i_{0}$ and since
$\mathbf{w}^{\prime}\in\sigma_{p^{i_{0}}\boldsymbol{\gamma}^{\prime}}$, we
have that
$p^{-i_{0}}A=(p^{-i_{0}}A)^{+}_{\boldsymbol{\gamma}^{\prime},\mathbf{w}}=(p^{-i_{0}}A)^{+}_{\boldsymbol{\gamma}^{\prime},\mathbf{w}^{\prime}}$.
Furthermore, for each $i>i_{0}$ and each $\mathbf{a}\in A$, we have
$\mathbf{w}^{\prime}\cdot(p^{-i}\mathbf{a})>\mathbf{w}^{\prime}\cdot(p^{-i_{0}}\mathbf{a})$.
Thus, for each $i>i_{0}$, we have
$(p^{-i}A)^{+}_{\boldsymbol{\gamma}^{\prime},\mathbf{w}^{\prime}}=p^{-i}A$ and
$(p^{-i}A)^{-}_{\boldsymbol{\gamma}^{\prime},\mathbf{w}^{\prime}}=\emptyset$.

Therefore, for all $i\geq 1$, we have
$(p^{-i}A)^{+}_{\boldsymbol{\gamma}^{\prime},\mathbf{w}}=(p^{-i}A)^{+}_{\boldsymbol{\gamma}^{\prime},\mathbf{w}^{\prime}}$
and
$(p^{-i}A)^{-}_{\boldsymbol{\gamma}^{\prime},\mathbf{w}}=(p^{-i}A)^{-}_{\boldsymbol{\gamma}^{\prime},\mathbf{w}^{\prime}}$.
It follows that
$S^{+}_{\boldsymbol{\gamma}^{\prime},\mathbf{w}}=S^{+}_{\boldsymbol{\gamma}^{\prime},\mathbf{w}^{\prime}}$
and
$S^{-}_{\boldsymbol{\gamma}^{\prime},\mathbf{w}}=S^{-}_{\boldsymbol{\gamma}^{\prime},\mathbf{w}^{\prime}}$.
∎

Let $\nu:K_{\mathbf{w}}\rightarrow\mathbb{R}\cup\\{\infty\\}$ be the valuation
defined by

$\nu(f):=\text{min}\\{\mathbf{w}\cdot\mathbf{a}:\mathbf{a}\in\text{supp}(f)\\}$

for $f\neq 0$ and $\nu(0):=\infty$.

######  Remark 3.3 (Properties of $K_{\mathbf{w}}$ with the valuation $\nu$).

  *   * (i)

The valued field $K_{\mathbf{w}}$ has valuation ring
${R_{\mathbf{w}}}:=\\{f\in K_{\mathbf{w}}:\nu(f)\geq 0\\}$. The valuation ring
has maximal ideal $\mathfrak{m}=\\{f\in K_{\mathbf{w}}:\nu(f)>0\\}$. The
residue field ${R_{\mathbf{w}}}/\mathfrak{m}$ is isomorphic to the field of
coefficients $\mathbbm{k}$; the map which sends
$\overline{f}\in{R_{\mathbf{w}}}/\mathfrak{m}$ to the constant term of
$f\in{R_{\mathbf{w}}}$ gives the isomorphism.

  * (ii)

The value group of $K_{\mathbf{w}}$ is $\mathbb{Q}^{d}$, which is
$n$-divisible for all $n$.

  * (iii)

By [Rayner1]*Theorem 2, $K_{\mathbf{w}}$ is a _Henselian valued field_. This
means that $R_{\mathbf{w}}$ satisfies Hensel’s lemma: if
$g(X)\in{R_{\mathbf{w}}}[X]$ and its reduction mod $\mathfrak{m}$,
$\overline{g}(X)\in\mathbbm{k}[X]$ has a simple root $a\in\mathbbm{k}$, then
there exists a unique $b\in R_{\mathbf{w}}$ such that $g(b)=0$ and
$\overline{b}=a\in\mathbbm{k}$.

We can now prove the main theorem of this section. Our proof is essentially
the same as [Saavedra, Theorem 5.3] except that the third bullet point in
Saavedra’s proof is replaced by our Lemma 3.2.

######  Theorem 3.4.

Fix an algebraically closed field $\mathbbm{k}$ of characteristic $p>0$. The
field $K_{\mathbf{w}}=\mathbbm{k}^{\mathbb{Q}^{d}}(\mathcal{A}_{\mathbf{w}})$
is algebraically closed.

###### Proof.

We proceed by contradiction and assume that $K_{\mathbf{w}}$ is not
algebraically closed. Then $K_{\mathbf{w}}$ admits a proper extension of
finite degree. By [Rayner1]*Lemma 4 there exists $f\in K_{\mathbf{w}}$ with
$\nu(f)<0$ such that the polynomial $X^{p}-X-f\in K_{\mathbf{w}}[X]$ is
irreducible. We express $f$ as the sum of two elements $f^{+},f^{-}\in
K_{\mathbf{w}}$ as follows:

$f^{+}(\mathbf{v})=\begin{cases}f(\mathbf{v})&\mathbf{v}\in\operatorname{supp}(f),\mathbf{w}\cdot\mathbf{v}\geq
0\\\ 0&\text{otherwise}\end{cases}\quad
f^{-}(\mathbf{v})=\begin{cases}f(\mathbf{v})&\mathbf{v}\in\operatorname{supp}(f),\mathbf{w}\cdot\mathbf{v}<0\\\
0&\text{otherwise.}\end{cases}$

Since $\operatorname{supp}(f^{+})$ and $\operatorname{supp}(f^{-})$ are
subsets of $\operatorname{supp}(f)\in\mathcal{A}_{\mathbf{w}}$, it follows
that
$\operatorname{supp}(f^{+}),\operatorname{supp}(f^{-})\in\mathcal{A}_{\mathbf{w}}$.
It suffices to prove the following two claims:

Claim 1. There exists $g\in K_{\mathbf{w}}$ that is a root of $X^{p}-X-f^{+}$.

Claim 2. There exists $h\in K_{\mathbf{w}}$ that is a root of $X^{p}-X-f^{-}$.

Indeed, for such $g$ and $h$ we have that $g+h$ is a root of $X^{p}-X-f$ which
contradicts the irreducibility of $X^{p}-X-f$. Hence $K_{\mathbf{w}}$ is
algebraically closed.

Proof of Claim 1. We have that $f^{+}$ belongs to the valuation ring
${R_{\mathbf{w}}}$ of the valued field $K_{\mathbf{w}}$, since $\nu(f^{+})\geq
0$ by definition. We will use the Henselian property of ${R_{\mathbf{w}}}$ to
find the root. Indeed, the reduction of $X^{p}-X-f^{+}$ modulo the maximal
ideal $\mathfrak{m}\subseteq{R_{\mathbf{w}}}$ is
$X^{p}-X-f^{+}(\mathbf{0})\in\mathbbm{k}[X]$. Because $\mathbbm{k}$ is
algebraically closed, $X^{p}-X-f^{+}(\mathbf{0})$ factors. Furthermore,
$X^{p}-X-f^{+}(\mathbf{0})$ has simple roots because its derivative is $-1$.
By the Henselian property each of these roots lifts to a distinct root of
$X^{p}-X-f^{+}$, so the desired root $g$ exists. This completes the proof of
Claim 1.

Proof of Claim 2. Define $h:\mathbb{Q}^{d}\to\mathbbm{k}$ by

$h(\mathbf{v})=\begin{cases}\sum_{j=1}^{\infty}\left(f^{-}(p^{j}\mathbf{v})\right)^{p^{-j}}&\mathbf{w}\cdot{\mathbf{v}}<0\\\
0&\text{ otherwise}.\end{cases}$

We will see in (5) that this is exactly the root we want.

_Subclaim a_. $h$ is well defined.

Since $\mathbbm{k}$ is algebraically closed of characteristic $p$, the
$p^{j}$th roots exist and are unique. We now show that the infinite sum in the
definition of $h$ is in fact a finite sum. Assume for contradiction that there
are infinitely many $p^{j}\mathbf{v}$s in $\operatorname{supp}(f^{-})$. Since
$\mathbf{w}\cdot\mathbf{v}<0$, these infinitely many $p^{j}\mathbf{v}$s form
an infinite strictly decreasing sequence. This is a contradiction:
$\operatorname{supp}(f^{-})\in\mathcal{A}_{\mathbf{w}}$ and so
$\operatorname{supp}(f^{-})$ is well ordered by item (i) of the definition of
field family (Definition 2.1).

_Subclaim b_. $\operatorname{supp}(h)\in\mathcal{A}_{\mathbf{w}}$.

Since $\operatorname{supp}(h)$ is a subset of
$S=\bigcup_{i=1}^{\infty}p^{-i}\operatorname{supp}(f)$, and
$S\in\mathcal{A}_{\mathbf{w}}$, by Lemma 3.2, we conclude that
$\operatorname{supp}(h)\in\mathcal{A}_{\mathbf{w}}$ as
$\mathcal{A}_{\mathbf{w}}$ is a field family.

_Subclaim c_. $h^{p}-h-f^{-}=0$.

First suppose $\mathbf{w}\cdot\mathbf{v}<0$. Then $\mathbf{w}\cdot
p^{-1}\mathbf{v}<0$. We claim that

$h^{p}(\mathbf{v})=h\left(p^{-1}\mathbf{v}\right)^{p}=\sum_{j=1}^{\infty}\left(f^{-}(p^{j-1}\mathbf{v})\right)^{p^{-(j-1)}}.$

The first equality is the Frobenius homomorphism, and the second equality is
by the definition of $h$. Hence, we have

(5) $\displaystyle h^{p}(\mathbf{v})-h(\mathbf{v})-f^{-}(\mathbf{v})$
$\displaystyle=\sum_{j=1}^{\infty}\left(f^{-}(p^{j-1}\mathbf{v})\right)^{p^{-(j-1)}}-\sum_{j=1}^{\infty}\left(f^{-}(p^{j}\mathbf{v})\right)^{p^{-j}}-f^{-}(\mathbf{v})$
$\displaystyle=\sum_{j^{\prime}=0}^{\infty}\left(f^{-}(p^{j^{\prime}}\mathbf{v})\right)^{p^{-j^{\prime}}}-\sum_{j=1}^{\infty}\left(f^{-}(p^{j}\mathbf{v})\right)^{p^{-j}}-f^{-}(\mathbf{v})$
$\displaystyle=f^{-}(\mathbf{v})-f^{-}(\mathbf{v})=0.$

On the other hand, when $\mathbf{w}\cdot\mathbf{v}\geq 0$ we have
$h^{p}(\mathbf{v})=h(\mathbf{v})=0=f^{-}(\mathbf{v})$ by definition. So $h$
satisfies the polynomial $X^{p}-X-f^{-}$. ∎

##  4\. Toric Bertini Theorems

In this section we prove the main theorem of this paper: Theorem 1.1.

If $\operatorname{char}(\mathbbm{k})>0$ set
$p=\operatorname{char}(\mathbbm{k})$; otherwise set $p=1$. For
$\mathbf{w}\in\mathbb{R}_{li}^{d}$, let $K_{\mathbf{w}}$ be the field of
$p$-discrete series in direction $\mathbf{w}$ with coefficients in
$\mathbbm{k}$, with variables $t_{1},\dots,t_{d}$, as defined in Definition
2.8. If $\operatorname{char}(\mathbbm{k})=0$, we set $p=1$ in this definition;
this is still an algebraically closed field by [McD, ArocaIlardi]. For an open
cone $C$ containing $\mathbf{w}$, let $K_{C}$ be the subring of
$K_{\mathbf{w}}$ consisting of those elements whose supports are well ordered
with respect to $\preceq_{\mathbf{w}^{\prime}}$ for every
$\mathbf{w}^{\prime}\in C\cap\mathbb{R}_{li}^{d}$, and for which the only
allowable denominators in exponents are powers of $p$. The fact that $K_{C}$
is a subring follows from the fact that unions and sums of well-ordered
subsets of $\mathbb{Q}^{d}$ are again well ordered. We will use the following
properties of $K_{C}$:

  1. (1)

For $\alpha\in K_{C}$, there is $\boldsymbol{\gamma}\in\mathbb{Q}^{d}$ with
$\operatorname{supp}(\alpha)\subset\boldsymbol{\gamma}+C^{\vee}$.

  2. (2)

Polynomials $\alpha\in\mathbbm{k}[t_{1}^{\pm 1},\dots,t_{d}^{\pm 1}]$ are in
$K_{C}$ for every cone $C$.

  3. (3)

If $C^{\prime}\subseteq C$, then $K_{C}$ is a subring of $K_{C^{\prime}}$.

Let $\mathbbm{k}\\{\\!\\{x\\}\\!\\}$ be the field of generalized Puiseux
series in one variable $x$. This consists of all formal power series with
coefficients in $\mathbbm{k}$ whose support is a well-ordered subset of
$\bigcup_{i=0}^{\infty}\frac{1}{Np^{i}}\mathbb{Z}$ for some integer $N$.

######  Definition 4.1.

Fix $\mathbf{w}\in\mathbb{R}^{d}_{li}$, and an open cone $C$ containing
$\mathbf{w}$. For $\mathbf{n}=(n_{1},\ldots,n_{d})\in C\cap\mathbb{Z}^{d}$,
and
$\boldsymbol{\theta}=(\theta_{1},\ldots,\theta_{d})\in(\mathbbm{k}^{\ast})^{d}$
we define
$\phi^{\boldsymbol{\theta}}_{\mathbf{n}}:K_{C}\rightarrow\mathbbm{k}\\{\\!\\{x\\}\\!\\}$
by

$\phi^{\boldsymbol{\theta}}_{\mathbf{n}}\left(\sum
c_{\mathbf{u}}\mathbf{t}^{\mathbf{u}}\right)=\sum
c_{\mathbf{u}}\boldsymbol{\theta}^{\mathbf{u}}x^{\mathbf{n}\cdot\mathbf{u}}.$

######  Lemma 4.2.

For every $\mathbf{n}=(n_{1},\ldots,n_{d})\in C\cap\mathbb{Z}^{d}$, and every
$\boldsymbol{\theta}\in(\mathbbm{k}^{*})^{d}$, the function
$\phi^{\boldsymbol{\theta}}_{\mathbf{n}}$ is well defined and is a ring
homomorphism from $K_{C}$ to $\mathbbm{k}\\{\\!\\{x\\}\\!\\}$.

###### Proof.

We first observe that the condition on $K_{C}$ that the denominators of
exponents be only powers of $p$ ensures that the expression
${\boldsymbol{\theta}}^{\mathbf{u}}$ is well defined, as $p$th roots are
unique in characteristic $p$. Fix $\alpha\in K_{C}$. To finish showing that
$\phi^{\boldsymbol{\theta}}_{\mathbf{n}}$ is well defined, we need to check
that the substitution does not map infinitely many terms of $\alpha$ to the
same term in $\mathbbm{k}\\{\\!\\{x\\}\\!\\}$. In other words, we need to
check that the fibers of the map

(6)
$\operatorname{supp}(\alpha)\rightarrow\mathbb{Q},\quad\mathbf{v}\mapsto\mathbf{n}\cdot\mathbf{v}$

are finite. Suppose on the contrary that for some $r\in\mathbb{Q}$ the
preimage
$\\{\mathbf{v}\in\operatorname{supp}(\alpha):\mathbf{n}\cdot\mathbf{v}=r\\}$
is infinite. Consider an infinite non-repeating sequence in this set.

Since $C$ is an open cone and $\mathbf{n}\in C$, there are vectors
$\mathbf{w}_{1},\dots,\mathbf{w}_{d}\in C\cap\mathbb{R}_{li}^{d}$ such that
$\mathbf{n}=a_{1}\mathbf{w}_{1}+\cdots+a_{d}\mathbf{w}_{d}$, for some positive
real numbers $a_{1},\dots,a_{d}$. Consider the infinite sequence above. Since
it is well ordered with respect to $\preceq_{\mathbf{w}_{1}}$, we can pass to
an increasing subsequence. We can now repeat this operation using the orders
given by $\mathbf{w}_{2},\dots,\mathbf{w}_{d}$, to get a sequence that is
increasing with respect to all the orders given by
$\mathbf{w}_{1},\dots,\mathbf{w}_{d}$. Thus the dot product of the sequence
with $\mathbf{n}$ must be strictly increasing as well, contradicting the
assumption that it is constant.

This shows that the substitution is a well-defined map. The fact that
$\phi^{\boldsymbol{\theta}}_{\mathbf{n}}$ is a ring homomorphism follows from
the fact that taking the dot product with $\mathbf{n}$ commutes with union and
sum of subsets of $\mathbb{Q}^{d}$. ∎

Since $\mathbbm{k}[t_{1},\dots,t_{d}]\subseteq K_{C}$ for all cones $C$, we
can also define an analogous homomorphism from
$\mathbbm{k}[t_{1},\dots,t_{d},y]$:

$\phi^{\boldsymbol{\theta}}_{\mathbf{n}}\colon\mathbbm{k}[t_{1},\dots,t_{d},y]\rightarrow\mathbbm{k}[x^{\pm
1},y]$

by

$\phi^{\boldsymbol{\theta}}_{\mathbf{n}}\left(\sum
c_{\mathbf{u},j}{\mathbf{t}}^{\mathbf{u}}y^{j}\right)=\sum\phi^{\boldsymbol{\theta}}_{\mathbf{n}}(c_{\mathbf{u},j}{\mathbf{t}}^{\mathbf{u}})y^{j}.$

We say an element $\alpha\in K_{C}$ has unbounded support if the support of
$\alpha$ is not contained in a bounded region in $\mathbb{R}^{d}$.

######  Lemma 4.3.

Let $\alpha\in K_{C}$ have unbounded support and be algebraic over
$\mathbbm{k}(t_{1},\dots,t_{d})$. Then there exists an open cone
$C^{\prime}\subseteq C$ containing $\mathbf{w}$ such that for any integer
vector $\mathbf{n}\in C^{\prime}$ and any
$\boldsymbol{\theta}\in(\mathbbm{k}^{*})^{d}$ the image
$\phi^{\boldsymbol{\theta}}_{\mathbf{n}}(\alpha)$ is not a polynomial in
$\mathbbm{k}[x^{\pm 1}]$.

###### Proof.

Since $\alpha$ is algebraic over $\mathbbm{k}(t_{1},\dots,t_{d})$, there is a
polynomial $h\in\mathbbm{k}[t_{1},\dots,t_{d},y]$ such that
$h(t_{1},\dots,t_{d},\alpha)=0$. Since
$\phi^{\boldsymbol{\theta}}_{\mathbf{n}}$ is a ring homomorphism for any
$\mathbf{n}\in C\cap\mathbb{Z}^{d}$ and any
$\boldsymbol{\theta}\in(\mathbbm{k}^{*})^{d}$, for such choices we then have

$\phi^{\boldsymbol{\theta}}_{\mathbf{n}}(h)(x,\phi^{\boldsymbol{\theta}}_{\mathbf{n}}(\alpha))=h(\theta_{1}x^{n_{1}},\ldots,\theta_{d}x^{n_{d}},\phi^{\boldsymbol{\theta}}_{\mathbf{n}}(\alpha))=0.$

Suppose $\phi^{\boldsymbol{\theta}}_{\mathbf{n}}(\alpha)$ is a polynomial in
$\mathbbm{k}[x^{\pm 1}]$. In order to have
$\phi^{\boldsymbol{\theta}}_{\mathbf{n}}(h)(x,\phi^{\boldsymbol{\theta}}_{\mathbf{n}}(\alpha))=0$,
there must be two distinct monomials $x^{a_{i}}y^{i}$ and $x^{a_{j}}y^{j}$ in
$\phi^{\boldsymbol{\theta}}_{\mathbf{n}}(h)$ that have the same maximal
$x$-degree after plugging in $\phi^{\boldsymbol{\theta}}_{\mathbf{n}}(\alpha)$
for $y$. Then

$a_{i}+i\cdot\deg_{x}\left(\phi^{\boldsymbol{\theta}}_{\mathbf{n}}(\alpha)\right)=a_{j}+j\cdot\deg_{x}\left(\phi^{\boldsymbol{\theta}}_{\mathbf{n}}(\alpha)\right)$

so $i\neq j$ and we have
$\deg_{x}\phi^{\boldsymbol{\theta}}_{\mathbf{n}}(\alpha)=\frac{a_{i}-a_{j}}{j-i}$.
Since each monomial in $\phi^{\boldsymbol{\theta}}_{\mathbf{n}}(h)$ is the
specialization of a monomial in $h$,

$\deg_{x}\phi^{\boldsymbol{\theta}}_{\mathbf{n}}(\alpha)\leq\max\left\\{\mathbf{n}\cdot\left(\frac{\mathbf{u}_{r}-\mathbf{u}_{s}}{s-r}\right):{\bf
t}^{\mathbf{u}_{r}}y^{r}\text{ and }{\bf t}^{\mathbf{u}_{s}}y^{s}\text{ are
monomials in }h\right\\}.$

For all directions $\mathbf{n}$ sufficiently close to $\mathbf{w}$, the
maximum is attained at the same pair of monomials ${\bf
t}^{\mathbf{u}_{i}}y^{i}$ and ${\bf t}^{\mathbf{u}_{j}}y^{j}$ in $h$.
Explicitly, we may take $\mathbf{n}$ in the open cone $C_{1}$ containing
$\mathbf{w}$ in the normal fan of the convex hull of the
$\frac{\mathbf{u}_{r}-\mathbf{u}_{s}}{s-r}$, which is also the normal cone of
$\frac{\mathbf{u}_{i}-\mathbf{u}_{j}}{j-i}$.

Since $\alpha\in K_{C}$, there is $\boldsymbol{\gamma}\in\mathbb{Q}^{d}$ with
$\operatorname{supp}(\alpha)\subset\boldsymbol{\gamma}+C^{\vee}$. Choose an
open cone $C_{2}\subseteq C_{1}\cap C$ whose closure is contained in the
interior of $C$.

Then
$(\boldsymbol{\gamma}+C^{\vee})\setminus(\frac{\mathbf{u}_{i}-\mathbf{u}_{j}}{j-i}+{C_{2}}^{\vee})$
is bounded, but $\operatorname{supp}(\alpha)$ is unbounded, so we can choose
$\mathbf{v}\in\operatorname{supp}(\alpha)\cap\frac{\mathbf{u}_{i}-\mathbf{u}_{j}}{j-i}+{C_{2}}^{\vee}$.
Then since
$\mathbf{v}\in\frac{\mathbf{u}_{i}-\mathbf{u}_{j}}{j-i}+{C_{2}}^{\vee}$, for
all $\mathbf{n}\in C_{2}$ we have
$\mathbf{n}\cdot\mathbf{v}\geq\mathbf{n}\cdot\frac{\mathbf{u}_{i}-\mathbf{u}_{j}}{j-i}$.
By axiom d of $p$-discreteness for $K_{\mathbf{w}}$, there is an open cone
$C_{3}$ containing $\mathbf{w}$ for which for any $\mathbf{n}\in
C_{3}\cap\mathbb{Z}^{d}$ the exponent $\mathbf{v}$ is the only preimage of the
map (6), so does not get cancelled after the substitution
$\phi^{\boldsymbol{\theta}}_{\mathbf{n}}$. Thus for $\mathbf{n}\in C_{2}\cap
C_{3}$ the monomial $x^{\mathbf{n}\cdot\mathbf{v}}$ appears in a term of
$\phi^{\boldsymbol{\theta}}_{\mathbf{n}}(\alpha)$, contradicting the degree
bound above. Thus $\phi^{\boldsymbol{\theta}}_{\mathbf{n}}(\alpha)$ cannot be
a polynomial for $\mathbf{n}\in C^{\prime}:=C_{2}\cap C_{3}$. ∎

######  Lemma 4.4.

Let $\alpha\in K_{C}$ be an element with infinite but bounded support. Then
there exists an open cone $C^{\prime}\subseteq C$ containing $\mathbf{w}$ and
a sublattice $H\subset\mathbb{Z}^{d}$ such that for any integer vector
$\mathbf{n}\in(C^{\prime}\setminus H)\cap\mathbb{Z}^{d}$, the substitution
$\phi^{\boldsymbol{\theta}}_{\mathbf{n}}(\alpha)$ is not a polynomial.
Moreover, the sublattice $H$ can be chosen to have an arbitrarily large index
in $\mathbb{Z}^{d}$.

###### Proof.

The support of $\alpha$ must contain points in $\mathbb{Q}^{d}$ with
coordinates whose denominators (in reduced form) are arbitrarily large because
there are only finitely many elements in $\mathbb{Q}^{d}$ in a bounded region
with denominators smaller than a given bound. Let $\mathbf{v}$ be an element
of $\operatorname{supp}(\alpha)$ with a non-integer coordinate. By axiom d of
$p$-discreteness, there is an open cone $C^{\prime}\subseteq C$ containing
$\mathbf{w}$ for which for any $\mathbf{n}\in C^{\prime}\cap\mathbb{Z}^{d}$
the term $\mathbf{v}$ is the only preimage of the map (6), so does not get
cancelled by the substitution $\phi^{\boldsymbol{\theta}}_{\mathbf{n}}$. Let
$H=\\{\mathbf{n}\in\mathbb{Z}^{d}\mid\mathbf{n}\cdot\mathbf{v}\in\mathbb{Z}\\}$.
For all integer vectors $\mathbf{n}\in C^{\prime}$ not in $H$, the map
$\phi_{\mathbf{n}}^{\boldsymbol{\theta}}$ sends the term $\mathbf{t^{v}}$ in
$\alpha$ to the term
$\boldsymbol{\theta}^{\mathbf{v}}x^{\mathbf{n}\cdot\mathbf{v}}$ in
$\phi^{\boldsymbol{\theta}}_{\mathbf{n}}(\alpha)$ with non-integer exponent,
so $\phi^{\boldsymbol{\theta}}_{\mathbf{n}}(\alpha)$ is not a polynomial. The
lattice $H$ is a sublattice of $\mathbb{Z}^{d}$ but is not all of
$\mathbb{Z}^{d}$ because if the denominator (in reduced form) of $j$th
coordinate of $\mathbf{v}$ is $M>1$, then
$\mathbf{e}_{j},2\mathbf{e}_{j},\dots,(M-1)\mathbf{e}_{j}$ are not in $H$, so
$H$ has index at least $M$ in $\mathbb{Z}^{d}$. The element
$\mathbf{v}\in\operatorname{supp}(\alpha)$ can be chosen to have a coordinate
with an arbitrarily large denominator, so the sublattice $H$ to avoid can be
chosen to have an arbitrarily large index. ∎

Following Amoroso-Sombra [amoroso2017factorization], we say that for an
irreducible variety $X$, a map $\pi:X\rightarrow(\mathbbm{k}^{*})^{d}$ has the
PB property  if for every isogeny (surjective group homomorphism with finite
kernel) $\lambda$ of $(\mathbbm{k}^{*})^{d}$ the pullback
$\lambda^{*}X:=X\times_{\lambda}(\mathbbm{k}^{*})^{d}$ in (7) is irreducible.

(7)
${X\times_{\lambda}(\mathbbm{k}^{*})^{d}}$${X}$${(\mathbbm{k}^{*})^{d}}$${(\mathbbm{k}^{*})^{d}}$$\scriptstyle{\lambda^{*}\pi}$$\scriptstyle{\pi}$$\scriptstyle{\lambda}$

######  Example 4.5.

Consider the variety $X=\operatorname{V}(x^{2}-yz^{2})$. The map
$\pi:X\rightarrow(\mathbbm{k}^{*})^{2}$ projecting onto the first two
coordinates does not satisfy the PB property, since for the isogeny
$\lambda:(\mathbbm{k}^{*})^{2}\rightarrow(\mathbbm{k}^{*})^{2}$ given by
$(x,y)\mapsto(x,y^{2})$ we have that
$X\times_{\lambda}(\mathbbm{k}^{*})^{2}=\operatorname{V}(x^{2}-y^{2}z^{2})=\operatorname{V}(x-yz)\cup\operatorname{V}(x+yz)$
is reducible.

######  Theorem 4.6.

Let $f\in\mathbbm{k}[t^{\pm 1}_{1},\dots,t^{\pm 1}_{d},y]$ be irreducible and
monic in $y$. Suppose that the projection of
$\operatorname{V}(f)\subseteq(\mathbbm{k}^{*})^{d}\times\mathbb{A}^{1}$ onto
the first $d$ coordinates has the PB property. Fix
$\mathbf{w}\in\mathbb{R}^{d}_{li}$. Then there exists an open cone $C$
containing $\mathbf{w}$ and finitely many sublattices
$H_{1},\dots,H_{r}\subset\mathbb{Z}^{d}$ such that for all vectors
$\mathbf{n}\in(C\cap\mathbb{Z}^{d})\setminus(H_{1}\cup\cdots\cup H_{r})$, and
all $\boldsymbol{\theta}\in(\mathbbm{k}^{*})^{d}$ the Laurent polynomial
$f(\theta_{1}x^{n_{1}},\dots,\theta_{d}x^{n_{d}},y)\in\mathbbm{k}[x^{\pm
1},y]$ is irreducible. Moreover, the sublattices can be chosen to be of
arbitrarily high index.

###### Proof.

Consider $f$ as a polynomial in $K_{\mathbf{w}}[y]$. Since $K_{\mathbf{w}}$ is
algebraically closed, we can write

$f=\prod_{i=1}^{s}(y-\alpha_{i})$

where $\alpha_{i}\in K_{\mathbf{w}}$. By the construction of $K_{\mathbf{w}}$,
there is $N>0$ for which every denominator of an exponent appearing in some
$\alpha_{i}$ can be put in the form $Np^{j}$ for some $j\geq 0$, where $p=1$
if $\operatorname{char}(\mathbbm{k})=0$. The isogeny
$\mu\colon(\mathbbm{k}^{*})^{d}\rightarrow(\mathbbm{k}^{*})^{d}$ given by
$t_{i}\mapsto t_{i}^{N}$ extends to an inclusion of fields $\mu\colon
K_{\mathbf{w}}\rightarrow K_{\mathbf{w}}$, and a map
$K_{\mathbf{w}}[y]\rightarrow K_{\mathbf{w}}[y]$, sending $y$ to $y$. We then
have $g:=\mu(f)=\prod_{i=1}^{s}(y-\mu(\alpha_{i}))$. It suffices to prove the
theorem for $g$, as
$\phi^{\boldsymbol{\theta}}_{\mathbf{n}}(g)=\phi_{N\mathbf{n}}^{\boldsymbol{\theta}}(f)$,
and if $\phi_{N\mathbf{n}}^{\boldsymbol{\theta}}(f)$ is irreducible, so is
$\phi^{\boldsymbol{\theta}}_{\mathbf{n}}(f)$. We thus henceforth assume that
$N=1$.

Since $N=1$, all denominators are powers of $p$, so by axioms a and b of
$p$-discreteness there is an open cone $C$ with $\alpha_{i}\in K_{C}$ for all
$1\leq i\leq s$. Thus for all $\mathbf{n}\in C\cap\mathbb{Z}^{d}$ and all
$\boldsymbol{\theta}\in(\mathbbm{k}^{*})^{d}$ we have

$\phi_{\mathbf{n}}^{\boldsymbol{\theta}}(f)=\prod_{i=1}^{s}(y-\phi_{\mathbf{n}}^{\boldsymbol{\theta}}(\alpha_{i})),$

so all roots of $\phi_{\mathbf{n}}^{\boldsymbol{\theta}}(f)$ have the form
$\phi_{\mathbf{n}}^{\boldsymbol{\theta}}(\alpha_{i})$ for a root $\alpha_{i}$
of $f$. Since $\phi^{\boldsymbol{\theta}}_{\mathbf{n}}(f)$ is monic, if it is
not irreducible, then it can be factored into monic polynomials, and monic
factors of $\phi^{\boldsymbol{\theta}}_{\mathbf{n}}(f)$ are images under
$\phi^{\boldsymbol{\theta}}_{\mathbf{n}}$ of monic factors of $f\in
K_{\mathbf{w}}[y]$. Thus we can consider all of the finitely many ways to
factor $f$ into two monic polynomials in $K_{\mathbf{w}}[y]$ and show that
“most” substitutions of $\mathbf{n}$ do not make the factors into polynomials
in $\mathbbm{k}[x^{\pm 1},y]$. Since $f$ has $\deg_{y}(f)$ roots in
$K_{\mathbf{w}}$, there are $m:=2^{\deg_{y}(f)-1}-1$ ways to factor $f$ into
two monic polynomials.

Since the projection $\pi:\operatorname{V}(f)\to(\mathbbm{k}^{*})^{d}$ onto
the first $d$ coordinates has the PB property, $f$ cannot be factored into
polynomials in $K_{\mathbf{w}}[y]$ whose coefficients in $K_{\mathbf{w}}$ have
finite support. If it did have such a factorization, then the isogeny
$\lambda:t_{i}\mapsto t_{i}^{N^{\prime}}$ would clear the common denominator
$N^{\prime}$ of the exponents of the coefficients in the factorization,
violating the PB property. If a factorization of $f$ in $K_{\mathbf{w}}[y]$
involves a coefficient with unbounded support, then by Lemma 4.3 there is an
open cone containing $\mathbf{w}$ such that any integer vector $\mathbf{n}$ in
this cone gives a substitution that is not a polynomial factorization. If a
factorization involves a coefficient whose support is infinite but bounded, by
Lemma 4.4 we can choose a sublattice of index greater than $m$ for
$\mathbf{n}$ to avoid. Since the union of $m$ lattices with index greater than
$m$ cannot cover all of $\mathbb{Z}^{d}$, the conclusion follows. ∎

######  Remark 4.7.

The use of the field $K_{\mathbf{w}}$, instead of one of the larger fields of
generalized Puiseux series such as the one constructed in [Saavedra], was
crucial for Theorem 4.6. The fact that the polynomial $f$ factors completely
is a consequence of the fact that $K_{\mathbf{w}}$ is algebraically closed
(Theorem 3.4), which used axiom d of Definition 2.2. The fact that the roots
of the transformed polynomial $g$ all live in $K_{C}$ for some open cone $C$
is a consequence of axiom a of Definition 2.2. Without this reduction, the
specialization homomorphism $\phi^{\boldsymbol{\theta}}_{\mathbf{n}}$ might
not be well defined.

We are now ready to prove the main theorem of this paper. Recall that a
morphism $\psi\colon(\mathbbm{k}^{*})^{r}\rightarrow(\mathbbm{k}^{*})^{d}$ is
given in coordinates by
$\psi(t_{1},\dots,t_{r})=(c_{1}\mathbf{t}^{\mathbf{a}_{1}},\dots,c_{d}\mathbf{t}^{\mathbf{a}_{d}})$,
where $c_{i}\in\mathbbm{k}^{*}$ and $\mathbf{a}_{i}\in\mathbb{Z}^{r}$ for
$1\leq i\leq d$, and $c_{i}=1$ for all $i$ if $\psi$ is an embedding of tori.
Let $A$ be the $r\times d$ matrix with columns the $\mathbf{a}_{i}$. The
morphism $\psi$ is an embedding if the matrix $A$ has rank $r$, and the
greatest common divisor of the $r\times r$ minors of $A$ is one. Changes of
coordinates on $(\mathbbm{k}^{*})^{r}$ correspond to row operations on $A$, so
we conclude that $r$-dimensional subtori of $(\mathbbm{k}^{*})^{d}$ correspond
to rational points in the Grassmannian $\operatorname{Gr}(r,d)$.

######  Theorem 1.1 (Toric Bertini).

Let $\mathbbm{k}$ be an algebraically closed field of arbitrary
characteristic. Let $X$ be a $d$-dimensional irreducible subvariety of
$(\mathbbm{k}^{*})^{n}$ where $d\geq 2$, and let
$\pi\colon(\mathbbm{k}^{*})^{n}\rightarrow(\mathbbm{k}^{*})^{d}$ be a morphism
with $\pi|_{X}$ dominant and finite. Suppose that $\pi|_{X}$ satisfies PB.
Then for every $1\leq r\leq d-1$ the set of $r$-dimensional subtori
$T\subseteq(\mathbbm{k}^{*})^{d}$ with $\pi^{-1}(\boldsymbol{\theta}\cdot
T)\cap X$ irreducible for all $\boldsymbol{\theta}\in(\mathbbm{k}^{*})^{d}$ is
dense in $\operatorname{Gr}(r,d)$.

###### Proof.

The morphism $\pi$ is given by
$\pi(\mathbf{t})_{j}=c_{j}\mathbf{t}^{\mathbf{p}_{j}}$ for $1\leq j\leq d$,
$c_{j}\in\mathbbm{k}^{*}$, and $\mathbf{p}_{j}\in\mathbb{Z}^{n}$. Let $P$ be
the $d\times n$ matrix with rows $\mathbf{p}_{1},\dots,\mathbf{p}_{d}$.
Integer row and column operations correspond to changes of coordinates on
$(\mathbbm{k}^{*})^{d}$ and $(\mathbbm{k}^{*})^{n}$, so we may assume that $P$
is in Smith normal form, and thus the morphism $\pi$ is projection onto the
first $d$ coordinates followed by an isogeny $\lambda\colon t_{i}\mapsto
t_{i}^{d_{i}}$ and multiplication by
$\mathbf{c}:=(c_{1},\dots,c_{d})\in(\mathbbm{k}^{*})^{d}$. Since $\pi|_{X}$
satisfies PB,
$X\times_{\lambda}(\mathbbm{k}^{*})^{d}\subset(\mathbbm{k}^{*})^{n}$ is
irreducible, and the map $\lambda^{*}\pi\colon
X\times_{\lambda}(\mathbbm{k}^{*})^{d}\rightarrow(\mathbbm{k}^{*})^{d}$ in (7)
is projection onto the first $d$ coordinates followed by multiplication by
$\mathbf{c}$. Furthermore, observe that $\lambda^{*}\pi$ satisfies PB. If
$T\subset(\mathbbm{k}^{*})^{d}$ is an $r$-dimensional subtorus with
$(\lambda^{*}\pi)^{-1}(\boldsymbol{\theta}\cdot T)$ irreducible for all
$\boldsymbol{\theta}\in(\mathbbm{k}^{*})^{d}$, then
$\pi|_{X}^{-1}(\lambda(\boldsymbol{\theta})\cdot T^{\prime})$ is irreducible
for $T^{\prime}=\lambda(T)$. It thus suffices to prove the theorem for
$X\times_{\lambda}(\mathbbm{k}^{*})^{d}$. As the factor $\mathbf{c}$ can be
absorbed into $\boldsymbol{\theta}$, we may thus assume that $\pi$ is
projection onto the first $d$ coordinates.

We now reduce to the case that $X$ is a hypersurface. Fix
$a_{1},\dots,a_{n}\in\mathbbm{k}$, and consider the morphism
$\rho\colon(\mathbbm{k}^{*})^{n}\rightarrow(\mathbbm{k}^{*})^{d}\times\mathbb{A}^{1}$
given by
$(t_{1},\dots,t_{n})\mapsto(t_{1},\dots,t_{d},\sum_{i=1}^{n}a_{i}t_{i})$.
Writing
$\pi^{\prime}\colon(\mathbbm{k}^{*})^{d}\times\mathbb{A}^{1}\rightarrow(\mathbbm{k}^{*})^{d}$
for the projection onto the first $d$ coordinates, we have the following
commuting diagram.

$\textstyle{X\subseteq(\mathbbm{k}^{*})^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\scriptstyle{\pi}$$\textstyle{(\mathbbm{k}^{*})^{d}\times\mathbb{A}^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi^{\prime}}$$\textstyle{(\mathbbm{k}^{*})^{d}}$

For generic $(a_{1},\dots,a_{n})$ the morphism $\rho$ is birational (for
example, by following the proof of [Hartshorne]*Proposition I.4.9; note that
this is independent of the characteristic of the field). Let $U\subset X$ be
an open set on which $\rho$ is an isomorphism, and let $Z=X\setminus U$. Since
$\pi$ is a proper morphism, as it is finite, we conclude that $\pi(Z)$ is a
subvariety of $(\mathbbm{k}^{*})^{d}$ of dimension at most $d-1$. Since $\rho$
is birational, the variety $\overline{\rho(X)}$ is $d$-dimensional, so is a
hypersurface in $(\mathbbm{k}^{*})^{d}\times\mathbb{A}^{1}$, defined by a
polynomial $f\in\mathbbm{k}[t_{1}^{\pm 1},\dots,t_{d}^{\pm 1},y]$. As
$\pi|_{X}$ is finite, the polynomial
$\sum_{i=1}^{n}a_{i}t_{i}\in\mathbbm{k}[X]$ satisfies a monic polynomial with
coefficients in $\mathbbm{k}[t_{1}^{\pm 1},\dots,t_{d}^{\pm 1}]$. This monic
polynomial must be a multiple of $f$, so $f$ is monic in $y$.

We next show that
$\pi^{\prime}:\operatorname{V}(f)\rightarrow(\mathbbm{k}^{*})^{d}$ satisfies
PB. Consider the Cartesian square

(8)
${\operatorname{V}(f)\times_{\mu}(\mathbbm{k}^{*})^{d}}$${\operatorname{V}(f)\subseteq(\mathbbm{k}^{*})^{d}\times\mathbb{A}^{1}}$${(\mathbbm{k}^{*})^{d}}$${(\mathbbm{k}^{*})^{d}}$$\scriptstyle{\alpha}$$\scriptstyle{\mu^{*}\pi^{\prime}}$$\scriptstyle{\pi^{\prime}}$$\scriptstyle{\mu}$

where $\mu$ is an isogeny. Since
$\operatorname{V}(f)\times_{\mu}(\mathbbm{k}^{*})^{d}$ is the hypersurface
defined by the specialization, determined by $\mu$, of the irreducible
polynomial $f$, all irreducible components of
$\operatorname{V}(f)\times_{\mu}(\mathbbm{k}^{*})^{d}$ are $d$-dimensional.
This, together with the finiteness of $\alpha$ (which follows since $\mu$ is
finite and finite morphisms are stable under base change), implies that all
irreducible components of
$\operatorname{V}(f)\times_{\mu}(\mathbbm{k}^{*})^{d}$ map by $\alpha$ onto
$\operatorname{V}(f)$.

Since $\rho|_{X}$ is birational, there exist non-empty open sets $U\subseteq
X$ and $W\subseteq\operatorname{V}(f)$ such that $\rho|_{U}:U\rightarrow W$ is
an isomorphism. Because all irreducible components of
$V(f)\times_{\mu}(\mathbbm{k}^{*})^{d}$ map by $\alpha$ onto the irreducible
$V(f)$, we have
$\overline{\alpha^{-1}(W)}=\operatorname{V}(f)\times_{\mu}(\mathbbm{k}^{*})^{d}$.
Thus, to see that $\operatorname{V}(f)\times_{\mu}(\mathbbm{k}^{*})^{d}$ is
irreducible, it suffices to show that
$\alpha^{-1}(W)=W\times_{\mu}(\mathbbm{k}^{*})^{d}$ is irreducible. Since the
isomorphism $\rho|_{U}:U\rightarrow W$ respects the maps to
$(\mathbbm{k}^{*})^{d}$, we have $W\times_{\mu}(\mathbbm{k}^{*})^{d}\cong
U\times_{\mu}(\mathbbm{k}^{*})^{d}$, so the irreducibility follows from the
fact that $\pi|_{X}$ satisfies $PB$. Thus $\pi^{\prime}$ satisfies PB.

We next observe that it suffices to prove the theorem for
$\pi^{\prime}\colon\operatorname{V}(f)\rightarrow(\mathbbm{k}^{*})^{d}$. Let
$T_{0}$ be any given $r$-dimensional subtorus of $(\mathbbm{k}^{*})^{d}$. Note
that for every $\boldsymbol{\theta}\in(\mathbbm{k}^{*})^{d}$ every irreducible
component $Y$ of $\pi|_{X}^{-1}(\boldsymbol{\theta}\cdot T_{0})$ satisfies
$\pi(Y)=\boldsymbol{\theta}\cdot T_{0}$. Indeed, since
$\boldsymbol{\theta}\cdot T_{0}$ is a complete intersection, by the Principal
Ideal Theorem (see for example, [Eisenbud]*Theorem 10.2) we have
$\dim(Y)\geq\dim(\boldsymbol{\theta}\cdot T_{0})$. Since $\pi$ is finite,
$\pi(Y)$ is a closed set, and $\dim(\pi(Y))=\dim(Y)$ (see for example,
[Eisenbud]*Proposition 9.2), so we conclude that
$\pi(Y)=\boldsymbol{\theta}\cdot T_{0}$. It thus follows that if $T_{0}$ is a
subtorus of $(\mathbbm{k}^{*})^{d}$ with $\boldsymbol{\theta}\cdot T_{0}$ not
contained in $\pi(Z)$ and $\pi|_{X}^{-1}(\boldsymbol{\theta}\cdot T_{0})$
reducible for some $\boldsymbol{\theta}\in(\mathbbm{k}^{*})^{d}$, then every
irreducible component of $\pi|_{X}^{-1}(\boldsymbol{\theta}\cdot T_{0})$
intersects $U$, so ${\pi^{\prime}}^{-1}(\boldsymbol{\theta}\cdot
T_{0})\cap\operatorname{V}(f)=\overline{\rho(\pi|_{X}^{-1}(\boldsymbol{\theta}\cdot
T_{0}))}$ is reducible as well.

To prove the theorem, it thus suffices to prove that we can choose an
$r$-dimensional subtorus $T$ arbitrarily close to $T_{0}$ in
$\operatorname{Gr}(r,d)$ with $\boldsymbol{\theta}\cdot T\not\subseteq\pi(Z)$,
and ${\pi^{\prime}}^{-1}(\boldsymbol{\theta}\cdot T)\cap\operatorname{V}(f)$
irreducible for all $\boldsymbol{\theta}\in(\mathbbm{k}^{*})^{d}$.

Note that if ${\pi^{\prime}}^{-1}(\boldsymbol{\theta}\cdot
T_{0})\cap\operatorname{V}(f)$ is reducible, then for any one-dimensional
subtorus $\tilde{T}\subset T_{0}$ the preimage
${\pi^{\prime}}^{-1}(\boldsymbol{\theta}\cdot\tilde{T})\cap\operatorname{V}(f)$
is also reducible if it is nonempty. To see this, consider a parameterization
of the torus $T_{0}$ by coordinates $z_{1},\dots,z_{r}$. Specializing $f$ to
this parameterization yields a reducible polynomial
$f^{\prime}\in\mathbbm{k}[z_{1}^{\pm 1},\dots,z_{r}^{\pm 1},y]$. The pullback
${\pi^{\prime}}^{-1}(\boldsymbol{\theta}\cdot\tilde{T})$ is a further nonzero
specialization of $f^{\prime}$, so remains reducible.

Let $A_{0}$ be an $r\times d$ matrix corresponding to $T_{0}$, and let
$\mathbf{w}\in\mathbb{R}^{d}_{li}$ be close to the first row of $A_{0}$ in the
Euclidean topology on $\mathbb{P}^{d-1}_{\mathbb{Q}}$ By Theorem 4.6, there is
an open cone $C$ containing $\mathbf{w}$ and a finite collection $\mathcal{H}$
of lattices of arbitrarily high index for which for any $\mathbf{n}\in C$ not
in any lattice in $\mathcal{H}$ the subtorus
$\tilde{T}=(t^{n_{1}},\dots,t^{n_{d}})$ has
${\pi^{\prime}}^{-1}(\boldsymbol{\theta}\cdot\tilde{T})\cap\operatorname{V}(f)$
irreducible for all $\boldsymbol{\theta}\in(\mathbbm{k}^{*})^{d}$. We assume
that the lower bound on the index of the lattices has been chosen to guarantee
that $\mathbb{Z}^{d}\setminus\bigcup_{H\in\mathcal{H}}H$ is nonempty. We have
$\boldsymbol{\theta}\cdot\tilde{T}\subseteq\pi(Z)$ only if the tropicalization
$\operatorname{trop}(\boldsymbol{\theta}\cdot\tilde{T})$, which is an affine
line in $\mathbb{R}^{d}$ with direction vector $\mathbf{n}$, is contained in
$\operatorname{trop}(\pi(Z))$. For generic $\mathbf{n}$, the polyhedral
complex $\operatorname{trop}(\pi(Z))$ does not contain any lines in the
direction $\mathbf{n}$.

Choose $\mathbf{n}\in C\setminus\bigcup_{H\in\mathcal{H}}H$ close to
$\mathbf{w}$ in the Euclidean topology on $\mathbb{P}^{d-1}_{\mathbb{Q}}$,
such that $\operatorname{trop}(\pi(Z))$ does not contain any lines in
direction $\mathbf{n}$, and let $A$ be the matrix obtained by replacing the
first row of $A_{0}$ by $\mathbf{n}$. Let $T$ be the $r$-dimensional subtorus
of $(\mathbbm{k}^{*})^{d}$ corresponding to $A$. This contains the one-
dimensional subtorus
$\tilde{T}=\\{(t^{n_{1}},\dots,t^{n_{d}}):t\in\mathbbm{k}^{*}\\}$, for which
${\pi^{\prime}}^{-1}(\boldsymbol{\theta}\cdot\tilde{T})\cap\operatorname{V}(f)$
is irreducible and $\boldsymbol{\theta}\cdot\tilde{T}\not\subseteq\pi(Z)$ for
all $\boldsymbol{\theta}\in(\mathbbm{k}^{*})^{d}$, so
${\pi^{\prime}}^{-1}(\boldsymbol{\theta}\cdot T)\cap\operatorname{V}(f)$ is
irreducible, and thus $\pi|_{X}^{-1}(\boldsymbol{\theta}\cdot T)$ is
irreducible for all such $\boldsymbol{\theta}$. As $T$ is close to $T_{0}$ in
$\operatorname{Gr}(r,d)$, this proves the theorem. ∎

######  Remark 4.8.

Theorem 1.1 differs from the version in [FMZ], in that we only show that the
set of exceptional tori are the complement of a dense set, rather than
essentially finite. As already discussed, the finiteness is not achievable
without the PB condition we, and [FMZ], impose. We do not know whether the PB
condition is necessary for the conclusion of Theorem 1.1. In our proof the
main use is to guarantee that the sublattices $H_{1},\dots,H_{r}$ to avoid do
not cover $\mathbb{Z}^{d}$, so the “genericity” condition for $\mathbf{n}$ is
nonempty. However we do not know an example of a projection $\pi$ failing PB
where the union of the lattices is all of $\mathbb{Z}^{d}$. In addition, it is
possible that the stronger finiteness conclusion of [FMZ] holds when we assume
PB. To prove this using our techniques, we would need a deeper understanding
of the structure of the cones $C$ used in Theorem 4.6. In characteristic zero,
under mild hypotheses, McDonald [McD] relates these cones to the normal fan of
the fiber polytope of a certain projection of the Newton polytope of the
polynomial $f$. It would be interesting to understand to what extent that can
be generalized to arbitrary characteristic.

##  5\. Tropical Bertini Theorem

One application of Theorem 1.1 is that the tropical Bertini theorem of
[maclagan2019higher] holds in arbitrary characteristic, and thus the
$d$-connectivity of the tropicalization of $d$-dimensional irreducible
varieties holds in arbitrary characteristic.

The _tropicalization_ of a variety $X\subseteq(\mathbbm{k}^{*})^{n}$, where
$\mathbbm{k}$ is a valued field with valuation
$\operatorname{val}\colon\mathbbm{k}^{*}\rightarrow\mathbb{R}$, is

$\operatorname{trop}(X)=\operatorname{cl}((\operatorname{val}(x_{1}),\dots,\operatorname{val}(x_{n})):(x_{1},\dots,x_{n})\in
X(L)),$

where $L/\mathbbm{k}$ is a nontrivially valued algebraically closed field
extension, and $\operatorname{cl}()$ is the closure in the usual Euclidean
topology on $\mathbb{R}^{n}$. By the Structure Theorem (see, for example,
[TropicalBook]*Chapter 3, or [BieriGroves]) $\operatorname{trop}(X)$ is the
support of a connected polyhedral complex.

In [maclagan2019higher] Maclagan and Yu proved a tropical Bertini theorem in
characteristic zero. We now remove the characteristic assumption.

We identify rational affine hyperplanes in $\mathbb{R}^{n}$ with points of
$\mathbb{P}^{n}_{\mathbb{Q}}$.

######  Theorem 1.2 (Tropical Bertini).

Let $X\subset(\mathbbm{k}^{*})^{n}$ be an irreducible $d$-dimensional variety,
with $d\geq 2$, over an algebraically closed field $\mathbbm{k}$ with
$\mathbb{Q}$ contained in the value group. The set of rational affine
hyperplanes $H$ in $\mathbb{R}^{n}$ for which the intersection
$\operatorname{trop}(X)\cap H$ is the tropicalization of an irreducible
variety is dense in the Euclidean topology on $\mathbb{P}^{n}_{\mathbb{Q}}$.

###### Proof.

We only need to replace a few sentences from the proof of
[maclagan2019higher]*Theorem 5. Explicitly, the proof holds essentially
verbatim (with the forward reference to Theorem 8 replaced by a forward
reference to our Theorem 1.1) until the sentence of
[maclagan2019higher]*Theorem 5 “We can now apply Theorem 8 to $\rho|_{Y}\colon
Y\rightarrow(\mathbbm{k}^{*})^{d}$”, where “Theorem 8” should again be
replaced by Theorem 1.1 of this paper. The proof then continues as follows.

By Theorem 1.1, since $\rho|_{Y}$ satisfies PB, and is dominant and finite,
the set of $(d-1)$-dimensional subtori
$T^{\prime}\subseteq(\mathbbm{k}^{*})^{d}$ such that
$\pi_{X}^{-1}(\boldsymbol{\theta}\cdot T^{\prime})$ is irreducible for all
$\boldsymbol{\theta}\in(\mathbbm{k}^{*})^{d}$ is dense in
$\operatorname{Gr}(d-1,d)\cong\mathbb{P}^{d-1}_{\mathbb{Q}}$. By definition,
this means that the set of affine hyperplanes
$\overline{H}=\operatorname{trop}(\boldsymbol{\theta}\cdot T^{\prime})$ for
such $\boldsymbol{\theta},T^{\prime}$ is dense in
$\mathbb{P}^{d}_{\mathbb{Q}}$.

Fix one such $\overline{H}=\operatorname{trop}(\boldsymbol{\theta}\cdot
T^{\prime})$, and let $H$ be the hyperplane in $\mathbb{R}^{n}$ defined by
$H=(\operatorname{trop}(\rho)^{-1}(\overline{H}))$. Since $\rho$ is a monomial
map, we have $H=\operatorname{trop}(\rho^{-1}(\boldsymbol{\theta}\cdot
T^{\prime}))$. As $\operatorname{trop}(\rho)$ is injective on every maximal
face of $\operatorname{trop}(Y)$, the intersection
$H\cap\operatorname{trop}(Y)$ is transverse, so by the Transverse Intersection
Lemma [OssermanPayne]*Theorem 1.1, [BJSST]*Lemma 15, [TropicalBook]*Theorem
3.4.12,

$H\cap\operatorname{trop}(Y)=\operatorname{trop}(\rho^{-1}|_{Y}(\boldsymbol{\theta}\cdot
T^{\prime})).$

Thus $H\cap\operatorname{trop}(Y)$ is the tropicalization of the irreducible
variety $\rho|_{Y}^{-1}(\boldsymbol{\theta}\cdot T^{\prime})$. Note that

$\operatorname{trop}(\mu^{\prime}(\rho|_{Y}^{-1}(\boldsymbol{\theta}\cdot
T^{\prime})))=\operatorname{trop}(\mu^{\prime})(H\cap\operatorname{trop}(Y))=\operatorname{trop}(\mu^{\prime})(H)\cap\operatorname{trop}(X)=\operatorname{trop}(\pi)^{-1}(\operatorname{trop}(\mu)(\overline{H})).$

Since $\operatorname{trop}(\mu)\in\operatorname{GL}(n,\mathbb{Q})$, and the
set of $\overline{H}$ is dense in $\mathbb{P}^{d}_{\mathbb{Q}}$, the set of
$\operatorname{trop}(\mu)(\overline{H})$ is dense in
$\mathbb{P}^{d}_{\mathbb{Q}}$ as required. ∎

In [maclagan2019higher] Theorem 1.2 is used to prove a higher connectivity
theorem for tropical varieties. While the statement of
[maclagan2019higher]*Theorem 1 assumes that the field $\mathbbm{k}$ has
characteristic zero, as observed in [maclagan2019higher]*Remark 11 this is
only needed to apply the tropical Bertini theorem, so in light of Theorem 1.2
we have the following generalization of [maclagan2019higher]*Theorem 1. A pure
polyhedral complex $\Sigma$ of dimension $d$ is $d$-connected through
codimension one if it is still connected after removing any $d-1$ closed
facets.

######  Theorem 5.1.

Let $\mathbbm{k}$ be a field that is either algebraically closed, complete, or
real closed with convex valuation ring. Let $X$ be a $d$-dimensional
irreducible subvariety of $(\mathbbm{k}^{*})^{n}$. Let $\Sigma$ be a pure
$d$-dimensional rational polyhedral complex with support
$|\Sigma|=\operatorname{trop}(X)$. Write $\ell$ for the dimension of the
lineality space of $\Sigma$. Then $\Sigma$ is $(d-\ell)$-connected through
codimension one.

## References

Authors’ addresses:

Department of Mathematics, Kalamazoo College, MI, USA
<EMAIL_ADDRESS>
School of Mathematics, The University of Edinburgh, Edinburgh EH9 3JZ, United
Kingdom<EMAIL_ADDRESS>
Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom
<EMAIL_ADDRESS>
Department of Mathematics: Algebra and Geometry, Ghent University, Belgium
<EMAIL_ADDRESS>
Department of Mathematics and Statistics, UiT - The Arctic University of
Norway, Tromsø, Norway

Department of Mathematics and Statistics, McMaster University, Canada
<EMAIL_ADDRESS>
School of Mathematics, Georgia Institute of Technology, Atlanta GA, USA
<EMAIL_ADDRESS>
School of Mathematics, Georgia Institute of Technology, Atlanta GA, USA
<EMAIL_ADDRESS>