# The asymptotic behavior of the Bergman kernel
on pseudoconvex model domains

Joe Kamimoto  Kyushu university, Motooka 744, Nishi-ku, Fukuoka, 819-0395,
Japan<EMAIL_ADDRESS>
###### Abstract.

In this paper, we investigate the asymptotic behavior of the Bergman kernel at
the boundary for some pseudoconvex model domains. This behavior can be
described by the geometrical information of the Newton polyhedron of the
defining function of the respective domains. We deal with not only the finite
type cases but also some infinite type cases.

##  1\. Introduction

Let $\Omega$ be a domain in ${\mathbb{C}}^{n}$ and let $A^{2}(\Omega)$ be the
Hilbert space of the $L^{2}$-holomorphic functions on $\Omega$. The Bergman
kernel $B_{\Omega}(z)$ of $\Omega$ (on the diagonal) is defined by
$B_{\Omega}(z)=\sum_{\alpha}|\phi_{\alpha}(z)|^{2}$, where
$\\{\phi_{\alpha}\\}$ is a complete orthonormal basis of $A^{2}(\Omega)$.
Throughout this paper, we assume that the boundary $\partial\Omega$ of
$\Omega$ is always $C^{\infty}$-smooth.

Since the behavior of the Bergman kernel at the boundary plays essentially
important roles in the study of several complex variables and complex
geometry, many interesting results about its behavior have been obtained.

In the case of strictly pseudoconvex domains, a beautiful asymptotic expansion
of the Bergman kernel was given by C. Fefferman [11] and Boutet de Monvel and
Sjöstrand [4]. In the case of weakly pseudoconvex domains, many kinds of
important results have been obtained (see the references in [15], [5], etc.).
In particular, in the two-dimensional finite type case, an asymptotic
expansion analogous to that of Fefferman was recently given by Hsiao and
Savale [14]. On the other hand, in the higher dimensional case, there does not
seem to be such strong and general results. In [15], the author investigated a
special case of pseudoconvex model domains and computed some asymptotic
expansion of the Bergman kernel. The purpose of this paper is to generalize
the results in [15].

In [15], only the finite type case was dealt with, while more general cases
will be considered in this paper; for example, some infinite type cases can be
also dealt with. Some two-dimensional infinite type cases have been precisely
investigated in [2], [3]. We will consider higher dimensional cases, which are
more complicated.

From the results in [15], [7], [6], [8], [5], it might be recognized that the
information of the boundary from the viewpoint of singularity theory is
valuable for the exact analysis of the Bergman kernel in the higher
dimensional weakly pseudoconvex case. In particular, the Newton polyhedron
determined from the boundary contains fruitful information for the singularity
of the Bergman kernel at the boundary.

One of the difficulty of the analysis in the infinite type case is caused by
the existence of non-zero flat functions (see Section 2.1). Notice that flat
functions do not affect the geometry of the Newton polyhedron and the
influence of flat functions is subtle in the singularity of the Bergman
kernel. However, this practical influence cannot always be negligible. In the
main theorem (Theorem 2), we will give a certain condition on the geometry of
the Newton polyhedron to determine the case where the above influence of flat
functions can be negligible in some sense.

This paper is organized as follows. In Section 2, we state a main theorem and
explain its significance. In Section 3, we exhibit an integral formula of the
Bergman kernel given by F. Haslinger [12], [13], on which our analysis is
based. In Section 4, we show that the singularity of the Bergman kernel can be
completely determined by the local geometry of the boundary. In our analysis,
it is necessary to consider various kinds of $C^{\infty}$ functions, but the
$C^{\infty}$ class contains many troublesome functions. In [18], [19], a
certain class of $C^{\infty}$ functions, called the $\hat{\mathcal{E}}$-class,
is introduced by the use of Newton polyhedra, which is easy to deal with.
Moreover, we also introduce a class analogous to the $\hat{\mathcal{E}}$-class
in the complex variables case in Section 5. In Section 6, we investigate the
singularity of the Bergman kernel in the $\hat{\mathcal{E}}$-case. In Section
7, a main theorem can be shown by the use of the results in the
$\hat{\mathcal{E}}$-case. By the way, the behavior of some Laplace type
integrals is a key in the analysis in Section 6. The work of Varchenko in [23]
(see also [1]) concerning local zeta functions and oscillatory integrals plays
crucial roles in the investigation of the above behavior. In Section 8, we
explain some results in [9], [18], [19], which generalize the above
Varchenko’s results, and apply these results to the anlaysis of the Bergman
kernel. In the last section, we will explain some important words and
concepts.

Notation and symbols.

  * •

We denote by ${\mathbb{N}}$, ${\mathbb{Z}}$, ${\mathbb{R}}$, ${\mathbb{C}}$
the set consisting of all natural numbers, integers, real numbers, complex
numbers, respectively. Moreover, we denote by ${\mathbb{Z}}_{+}$,
${\mathbb{R}}_{+}$ the set consisting of all nonnegative integers, real
numbers, respectively. For $s\in{\mathbb{C}}$, ${\rm Im}(s)$ expresses the
imaginary part of $s$.

  * •

Let $\alpha:=(\alpha_{1},\ldots,\alpha_{n})$,
$\beta:=(\beta_{1},\ldots,\beta_{n})\in{\mathbb{Z}}_{+}^{n}$. The multi-
indeces will be used as follows. For
$x=(x_{1},\ldots,x_{n})\in{\mathbb{R}}^{n}$, define

$\displaystyle x^{\alpha}=x_{1}^{\alpha_{1}}\cdots
x_{n}^{\alpha_{n}},\quad\|x\|_{{\mathbb{R}}}=\sqrt{x_{1}^{2}+\cdots+x_{n}^{2}}.$

For
$z=(z_{1},\ldots,z_{n}),\,\,\bar{z}=(\bar{z}_{1},\ldots,\bar{z}_{n})\in{\mathbb{C}}^{n}$,
define

$\displaystyle z^{\alpha}:=z_{1}^{\alpha_{1}}\cdots
z_{n}^{\alpha_{n}},\,\,\bar{z}^{\beta}:=\bar{z}_{1}^{\beta_{1}}\ldots\bar{z}_{n}^{\beta_{n}},\,\,|z|^{2\alpha}:=|z_{1}|^{2\alpha_{1}}\cdots|z_{n}|^{2\alpha_{n}},$
$\displaystyle\|z\|=\sqrt{|z_{1}|^{2}+\cdots+|z_{n}|^{2}}.$

  * •

For a positive number $R$, we denote

$\displaystyle
D_{{\mathbb{R}}}(R)=\\{x\in{\mathbb{R}}^{n}:\|x\|_{{\mathbb{R}}}<R\\},\quad
D(R)=\\{z\in{\mathbb{C}}^{n}:\|z\|<R\\}.$

##  2\. Main results

###  2.1. Newton data

In this paper, many concepts in convex geometry play useful roles. We will
explain the exact meanings of necessary words in convex geometry in Section
9.2 (see also [24]).

Let $F$ be a real-valued $C^{\infty}$ function defined near the origin in
${\mathbb{C}}^{n}$. Let

$\sum_{\alpha,\beta\in{\mathbb{Z}}_{+}^{n}}C_{\alpha\beta}z^{\alpha}\bar{z}^{\beta}=\sum_{\alpha,\beta\in{\mathbb{Z}}_{+}^{n}}C_{\alpha\beta}z_{1}^{\alpha_{1}}\cdots
z_{n}^{\alpha_{n}}\bar{z}_{1}^{\beta_{1}}\cdots\bar{z}_{n}^{\beta_{n}}$

be the Taylor series of $F$ at the origin. The support of $F$ is the set
$S_{F}=\\{\alpha+\beta\in{\mathbb{Z}}_{+}^{n}:C_{\alpha\beta}\neq 0\\}$ and
the Newton polyhedron of  $F$ is the integral polyhedron

${\mathcal{N}}_{+}(F)=\mbox{ the convex hull of the set
$\displaystyle\bigcup\\{\alpha+\beta+{\mathbb{R}}_{+}^{n}:\alpha+\beta\in
S_{F}\\}$ in ${\mathbb{R}}_{+}^{n}$.}$

We say that $F$ is flat if ${\mathcal{N}}_{+}(F)=\emptyset$ and that $F$ is
convenient if ${\mathcal{N}}_{+}(F)$ intersects all the axes. For a compact
face $\gamma$ of ${\mathcal{N}}_{+}(F)$, the $\gamma$-part of $F$ is defined
by

$F_{\gamma}(z)=\sum_{\alpha+\beta\in\gamma\cap{\mathbb{Z}}_{+}^{n}}C_{\alpha\beta}z^{\alpha}\bar{z}^{\beta}.$

We define a quantity $\rho_{F}$ ($\in{\mathbb{Z}}_{+}$) as follows. When $F$
is convenient, let

$\rho_{F}:=\max\\{\rho_{j}(F):j=1,\ldots,n\\},$

where

$\rho_{j}(F):=\min\\{t\geq
0:(0,\ldots,\stackrel{{\scriptstyle(j)}}{{t}},\ldots,0)\in{\mathcal{N}}_{+}(F)\\}.$

When $F$ is not convenient, let $\rho_{F}:=\infty$.

Hereafter, we assume that $F$ is not flat. The Newton distance of $F$ is the
nonnegative number

$d_{F}:=\min\\{t\geq 0:(t,\ldots,t)\in{\mathcal{N}}_{+}(F)\\}.$

Since $F$ is not flat, there exists the minimum proper face of the Newton
polyhedron ${\mathcal{N}}_{+}(F)$ containing the point
$P_{F}:=(d_{F},\ldots,d_{F})$, which is called the principal face of
${\mathcal{N}}_{+}(F)$ and is denoted by $\gamma_{*}$. The codimension of
$\gamma_{*}$ is called the Newton multiplicity of $F$, which is denoted by
$m_{F}$ (i.e., $m_{F}=n-\dim(\gamma_{*})$). In particular, when $P_{F}$ is a
vertex of ${\mathcal{N}}_{+}(F)$, $\gamma_{*}$ is the point $P_{F}$ and
$m_{F}=n$. When $\gamma_{*}$ is compact, the principal part of $F$ is defined
by

$F_{*}(z)=\sum_{\alpha,\beta\in\gamma_{*}\cap{\mathbb{Z}}_{+}^{n}}C_{\alpha\beta}z^{\alpha}\bar{z}^{\beta}$

(i.e., the principal part of $F$ is the $\gamma_{*}$-part of $F$).

###  2.2. Main results

Let $U$ be a complete pseudoconvex Reinhardt domain in ${\mathbb{C}}^{n}$
(possibly $U={\mathbb{C}}^{n}$). Let $F$ be a real-valued $C^{\infty}$
function on $U$ satisfying the following conditions

  1. (A)

$F(z)=0$ if and only if $z=0$ and $F$ is not flat at the origin;

  2. (B)

$F$ is a plurisubharmonic function on $U$;

  3. (C)

$F(e^{i\theta_{1}}z_{1},\ldots,e^{i\theta_{n}}z_{n})=F(z_{1},\ldots,z_{n})$
for any $\theta_{j}\in{\mathbb{R}}$ and $z\in U$;

  4. (D)

If $U$ is unbounded, then there are some positive numbers $c$, $\beta$, $L$
such that $F(z)\geq c\|z\|^{\beta}$ for $z\in U\setminus D(L)$.

We will mainly deal with pseudoconvex model domains in ${\mathbb{C}}^{n+1}$ of
the form

$\Omega_{F}=\\{(z_{0},z_{1},\ldots,z_{n})\in{\mathbb{C}}\times U:{\rm
Im}(z_{0})>F(z_{1},\ldots,z_{n})\\}.$

Note that the condition (D) implies that the dimension of $A^{2}(\Omega_{F})$
is infinity.

In the case of the domain $\Omega_{F}$, the finite type condition can be
easily seen in the information of the Newton polyhedron of $F$. Let
$\Delta_{1}(\partial\Omega_{F},0)$ be the D’Angelo type of
$\partial\Omega_{F}$ at the origin.

######  Lemma 1 ([16]).

$\Delta_{1}(\partial\Omega_{F},0)=\rho_{F}$. In particular, the following two
conditions are equivalent.

  1. (i)

$\partial\Omega_{F}$ is of finite type at $0$ (i.e.,
$\Delta_{1}(\partial\Omega_{F},0)<\infty$);

  2. (ii)

$F$ is convenient (i.e., ${\mathcal{N}}_{+}(F)$ intersects all axes).

Let $B_{\Omega_{F}}(z_{0},z)=B_{\Omega_{F}}(z_{0},z_{1},\ldots,z_{n})$ be the
Bergman kernel (on the diagonal) of $\Omega_{F}$. Since we are interested in
the behavior of the restriction of the Bergman kernel of $\Omega_{F}$ to the
vertical line, we define

${\mathcal{B}}_{F}(\rho):=B_{\Omega_{F}}(0+i\rho,0,\ldots,0)\quad\mbox{ for
$\rho>0$.}$

The behavior of ${\mathcal{B}}_{F}(\rho)$, as $\rho$ tends to $0$, can be
exactly expressed by the use of the geometry of the Newton polyhedron
${\mathcal{N}}_{+}(F)$ of $F$.

######  Theorem 1 ([15]).

If $\partial\Omega_{F}$ is of finite type at $0$, then

(1)
${\mathcal{B}}_{F}(\rho)=\frac{\Psi(\rho)}{\rho^{2/d_{F}+2}(\log\rho)^{m_{F}-1}},$

where $\Psi(\rho)$ admits the asymptotic expansion:

(2)
$\Psi(\rho)\sim\sum_{j=0}^{\infty}\sum_{k=a_{j}}^{\infty}C_{jk}\frac{\rho^{j/m}}{(\log(1/\rho))^{k}}\quad\mbox{as
$\rho\to 0$,}$

where $m$ is a positive integer, $a_{j}$ are integers and $C_{jk}$ are real
numbers (the exact meaning of the above asymptotic expansion is explained in
Section 9.1, below).

Furthermore, the first coefficient of the above expansion can be determined by
the use of the Newton data of $F$ as follows:

(3)
${\mathcal{B}}_{F}(\rho)=\frac{C(F_{*})}{\rho^{2/d_{F}+2}(\log\rho)^{m_{F}-1}}\cdot(1+o(\rho^{\varepsilon}))\quad\mbox{as
$\rho\to 0$,}$

where $\varepsilon$ is a positive constant and $C(F_{*})$ is a positive
constant depending only on the principal part $F_{*}$ of $F$.

In this paper, we improve the above theorem in some sense as follows.

######  Theorem 2.

Suppose that there exists a $C^{\infty}$ function $F_{0}$ defined near the
origin in ${\mathbb{C}}^{n}$ such that $F_{0}$ satisfies the condition (C) and
the $\hat{\mathcal{E}}$-condition at the origin (see Section 5) and
$F(z)-F_{0}(z)$ is a nonnegative flat function. If the principal face
$\gamma_{*}$ of the Newton polyhedron of $F$ is compact, then

(4)
${\mathcal{B}}_{F}(\rho)=\frac{C(F_{*})}{\rho^{2/d_{F}+2}(\log\rho)^{m_{F}-1}}\cdot(1+o(\rho^{\varepsilon}))\quad\mbox{as
$\rho\to 0$,}$

where $\varepsilon$ is a positive constant and $C(F_{*})$ is a positive
constant depending only on the principal part $F_{*}$ of $F$.

######  Remark .

(1) Note that the finite type condition is equivalent to the convenience
condition of $F$ from Lemma 1. When $F$ is convenient, $F$ itself satisfies
the $\hat{\mathcal{E}}$-condition (see [18], [19]) and the principal face of
${\mathcal{N}}_{+}(F)$ is always compact. Therefore, the assumption of Theorem
2 is weaker than that of Theorem 1. Indeed, Theorem 2 can be applied to some
infinite type cases.

(2) In the two-dimensional case (i.e. $\Omega_{F}\subset{\mathbb{C}}^{2}$),
Lemma 1 implies that the finite type condition is equivalent to the
nonflatness of $F$. Since we assume the nonflatness of $F$ in (A), the
advantage of Theorem 2 can be only seen in the case where the dimension is
higher than two.

(3) Theorem 2 can be applied to the following examples, which are in the
infinite type case.

  * •

When $F(z_{1},z_{2})=|z_{1}|^{6}+|z_{1}|^{2}|z_{2}|^{4}+e^{-1/|z_{2}|^{2}}$
near the origin,

${\mathcal{B}}_{F}(\rho)=\frac{C(F_{*})}{\rho^{8/3}}\cdot(1+o(\rho^{\varepsilon}))\quad\mbox{as
$\rho\to 0$.}$

(Note that $F_{*}(z_{1},z_{2})=|z_{1}|^{6}+|z_{1}|^{2}|z_{2}|^{4}$.)

  * •

When $F(z_{1},z_{2})=|z_{1}|^{6}+|z_{1}|^{2}|z_{2}|^{2}+e^{-1/|z_{2}|^{2}}$
near the origin,

${\mathcal{B}}_{F}(\rho)=\frac{C(F_{*})}{\rho^{3}\log\rho}\cdot(1+o(\rho^{\varepsilon}))\quad\mbox{as
$\rho\to 0$.}$

(Note that $F_{*}(z_{1},z_{2})=|z_{1}|^{2}|z_{2}|^{2}$.)

  * •

When
$F(z_{1},z_{2})=|z_{1}|^{2}|z_{2}|^{2}+e^{-1/|z_{1}|^{2}}+e^{-1/|z_{2}|^{2}}$
near the origin,

${\mathcal{B}}_{F}(\rho)=\frac{C(F_{*})}{\rho^{3}\log\rho}\cdot(1+o(\rho^{\varepsilon}))\quad\mbox{as
$\rho\to 0$.}$

(Note that $F_{*}(z_{1},z_{2})=|z_{1}|^{2}|z_{2}|^{2}$.)

  * •

When
$F(z_{1},z_{2},z_{3})=|z_{1}|^{8}+|z_{2}|^{8}+|z_{1}|^{2}|z_{2}|^{2}|z_{3}|^{2}+e^{-1/|z_{3}|^{2}}$
near the origin,

${\mathcal{B}}_{F}(\rho)=\frac{C(F_{*})}{\rho^{3}(\log\rho)^{2}}\cdot(1+o(\rho^{\varepsilon}))\quad\mbox{as
$\rho\to 0$.}$

(Note that $F_{*}(z_{1},z_{2})=|z_{1}|^{2}|z_{2}|^{2}|z_{3}|^{2}$.)

We remark that $e^{-1/|z_{j}|^{2}}$ ($j=1,2,3$) in the above examples can be
replaced by any flat functions which are positive away from the origin.

(4) Our method in the proof of Theorem 2 cannot generally show the existence
of an asymptotic expansion of the form (2). We guess that the pattern of the
asymptotic expansion might not be expressed as in the form (2) in general from
some observation of the strange phenomena in [20], [21], [22], which are seen
in the analytic continuation of local zeta functions.

(5) In the case where the principal face is noncompact, there exist many
examples in which the behavior of the Bergman kernel ${\mathcal{B}}_{F}(\rho)$
as $\rho\to 0$ is different from (4). For example, in the case where
$F(z_{1},z_{2})=|z_{1}|^{2}+e^{-1/|z_{2}|^{p}}$ near the origin,
${\mathcal{B}}_{F}(\rho)$ locally satisfies

$\frac{c_{1}|\log\rho|^{1/p}}{\rho^{3}}\leq{\mathcal{B}}_{F}(\rho)\leq\frac{c_{2}|\log\rho|^{1/p}}{\rho^{3}}$

near $\rho=0$, where $c_{1},c_{2}$ are positive constants ([17]). Notice that
the logarithmic functions appear in the numerators in the above estimates.
Observing the above estimates, we can see that the behavior of
${\mathcal{B}}_{F}(\rho)$ depends on $p$, which is an information of the flat
term $e^{-1/|z_{2}|^{p}}$. In the noncompact principal face case, the
information of the Newton polyhedron of $F$ cannot always determine the
singularity of the Bergman kernel completely.

(6) Of course, the constant $\varepsilon$ in (3), (4) does not depend on
$\rho$. The equations of similar type to (3), (4) will be often seen in this
paper. In these equations, the constant $\varepsilon$ can be chosen by the use
of the geometry of the respective Newton polyhedron.

Since $\rho^{\varepsilon}=o((\log(1/\rho)^{-1})$ holds for any
${\varepsilon}>0$, $o(\rho^{\varepsilon})$ in the equations in (3), (4) and in
the examples in Remark 1 (3) can be replaced by $o((\log(1/\rho)^{-1})$.

(7) It is desirable to show that the conditions (A-C) of $F$ imply the
existence of $F_{0}$ in the assumption of the theorem (in other words, the
existence of $F_{0}$ might be removed in the assumption). Even if the
existence of $F_{0}$ is not known, we can give the following estimate:

${\mathcal{B}}_{F}(\rho)\leq\frac{C(F_{*})}{\rho^{2/d_{F}+2}(\log\rho)^{m_{F}-1}}\cdot(1+C\rho^{\varepsilon})\quad\mbox{for
$\rho\in(0,\delta)$,}$

where $C(F_{*})$ is as in the theorem and $C,\varepsilon,\delta$ are positive
constants. This can be easily seen from the proof of Theorem 2.

##  3\. Integral formula of the Bergman kernel

Let $U$ be a domain in ${\mathbb{C}}^{n}$ and let $F:U\to{\mathbb{R}}_{+}$ be
a $C^{\infty}$-smooth plurisubharmonic function. The weighted Hilbert space
$H_{\tau}(U)$ ($\tau>0$) consists of all entire functions
$\psi:U\to{\mathbb{C}}$ such that

$\int_{U}|\psi(z)|^{2}e^{-2\tau F(z)}dV(z)<\infty,$

where $dV(z)$ denotes the Lebesgue measure on ${\mathbb{C}}^{n}$. We only
consider the case where $H_{\tau}(U)$ is a nontrivial Hilbert space with the
reproducing kernel. (When $U$ is bounded, this is obvious. When $U$ is
unbounded, the condition (D) in Section 2.2 can imply the above
nontriviality.) We denote the above kernel (on the diagonal) by
$K_{F}(z;\tau)$. We remark that the function $\tau\mapsto K_{F}(z;\tau)$ is
continuous for fixed $z\in U$ from the result in [10]. F. Haslinger [12], [13]
shows that the Bergman kernel $B_{\Omega_{F}}(z_{0},z)$ of the domain
$\Omega_{F}$ can be expressed by the use of $K_{F}(z;\tau)$ as follows.

(5)
$B_{\Omega_{F}}(z_{0},z)=\frac{1}{2\pi}\int_{0}^{\infty}e^{-2\rho\tau}K_{F}(z;\tau)\tau
d\tau,$

where $\rho$ is the imaginary part of $z_{0}$.

In the case where $F$ safisfies the conditions (C), (D) in Section 2.2, we can
take a complete orthonormal system for $H_{\tau}(U)$ as

$\left\\{\frac{z^{\alpha}}{c_{\alpha}(\tau)}:\alpha\in{\mathbb{Z}}_{+}^{n}\right\\},\quad\mbox{
with }c_{\alpha}(\tau)^{2}=\int_{U}|z|^{2\alpha}e^{-2\tau F(z)}dV(z).$

Therefore, $K_{F}(z;\tau)$ can be expressed as

(6)
$K_{F}(z;\tau)=\sum_{\alpha\in{\mathbb{Z}}_{+}^{n}}\frac{|z|^{2\alpha}}{c_{\alpha}(\tau)^{2}}.$

From (5), (6), ${\mathcal{B}}_{F}(\rho)$ can be expressed as

(7)
${\mathcal{B}}_{F}(\rho)=\frac{1}{2\pi}\int_{0}^{\infty}e^{-2\rho\tau}\frac{\tau}{{c}_{0}(\tau)^{2}}d\tau.$

In order to see the behavior of ${\mathcal{B}}_{F}(\rho)$ as $\rho\to 0$, it
suffices to investigate that of ${c}_{0}(\tau)^{2}$ as $\tau\to\infty$.

##  4\. Localization lemma

Let $U$ be an open neighborhood of the origin in ${\mathbb{C}}^{n}$ and let
$F:U\to{\mathbb{R}}$ be a nonnegative $C^{\infty}$ function with $F(0)=0$.

Let $R$ be a positive number such that $D(R)\subset U$. Let
$\varphi:{\mathbb{R}}^{n}\to[0,1]$ be a $C^{\infty}$ function such that
$\varphi$ identically equals $1$ on $D_{{\mathbb{R}}}(R/2)$ and its support is
contained in $D_{{\mathbb{R}}}(R)$. Let $\Phi:{\mathbb{C}}^{n}\to[0,1]$ be a
$C^{\infty}$ function defined by
$\Phi(z_{1},\ldots,z_{n})=\varphi(|z_{1}|,\ldots,|z_{n}|)$.

We define an integral of the form

(8)
$\tilde{\mathcal{B}}_{F}(\rho)=\frac{1}{2\pi}\int_{0}^{\infty}e^{-2\rho\tau}\frac{\tau}{\tilde{c}_{0}(\tau)^{2}}d\tau\quad\mbox{
for $\rho>0$},$

where

(9) $\tilde{c}_{0}(\tau)^{2}=\int_{U}e^{-2\tau F(z)}\Phi(z)dV(z).$

Since there exists a positive number $a$ such that $F(z)\leq a\|z\|$ on
$D(R)$, we see

(10) $\begin{split}\tilde{c}_{0}(\tau)^{2}&\geq\int_{D(R/2)}e^{-2\tau
F(z)}dV(z)\\\ &\geq c\int_{0}^{R/2}e^{-2a\tau
x}x^{2n-1}dx\geq\frac{c}{(2a\tau)^{2n}}\int_{0}^{1}e^{-s}s^{2n-1}ds=\frac{C}{\tau^{2n}},\end{split}$

for $\tau\geq 1$, where $c,C$ are positive constants independent of $\tau$.

By the use of the estimate (10), the integral $\tilde{\mathcal{B}}_{F}(\rho)$
in (8) can be considered as a $C^{\infty}$ function defined on $(0,\infty)$,
which is samely denoted by $\tilde{\mathcal{B}}_{F}(\rho)$.

Since $\tilde{c}_{0}(\tau)^{2}\leq c_{0}(\tau)^{2}$, the relationship between
$\tilde{\mathcal{B}}_{F}(\rho)$ and the Bergman kernel
${\mathcal{B}}_{F}(\rho)$ can be seen:
$\tilde{\mathcal{B}}_{F}(\rho)\geq{\mathcal{B}}_{F}(\rho)$. The following
proposition shows that the singularities of ${\mathcal{B}}_{F}(\rho)$ and
$\tilde{\mathcal{B}}_{F}(\rho)$ at the origin are essentially the same.

######  Proposition 1.

If $F$ satisfies the conditions (A–D) in Section 2.2, then
$\tilde{\mathcal{B}}_{F}(\rho)-{\mathcal{B}}_{F}(\rho)$ can be real-
analytically extended to an open neighborhood of $\rho=0$.

###### Proof.

From the integral expressions in (7) and (8), we have

${\mathcal{B}}_{F}(\rho)-\tilde{\mathcal{B}}_{F}(\rho)=\frac{1}{2\pi}\int_{0}^{\infty}e^{-2\rho\tau}\left(\frac{1}{c_{0}(\tau)^{2}}-\frac{1}{\tilde{c}_{0}(\tau)^{2}}\right)\tau
d\tau.$

Therefore, the proposition can be easily seen by the use of the following
lemma. ∎

######  Lemma 2.

There exist positive numbers $L,C,q,\epsilon$ such that

$\left|\frac{1}{\tilde{c}_{0}(\tau)^{2}}-\frac{1}{{c}_{0}(\tau)^{2}}\right|\leq
C\tau^{q}e^{-2\epsilon\tau}\quad\mbox{ for $\tau\geq L$.}$

###### Proof.

Let

$e(\tau):={c}_{0}(\tau)^{2}-\tilde{c}_{0}(\tau)^{2}=\int_{U}e^{-2\tau
F(z)}(1-\Phi(z))dV(z).$

From the conditions (A), (C) in Section 2.2, there exist positive numbers
$c,\epsilon$ such that

$F(z)\geq\epsilon+c(|z_{1}|^{\beta}+\cdots+|z_{n}|^{\beta})\quad\mbox{for
$z\in U\setminus D(R/2)$}.$

By the use of the above inequality, we can

(11) $\begin{split}|e(\tau)|&\leq\int_{U\setminus D(R/2)}e^{-2\tau
F(z)}dV(z)\\\
&\leq(2\pi)^{n}e^{-2\epsilon\tau}\int_{{\mathbb{R}}_{+}^{n}}e^{-2c\tau(x_{1}^{\beta}+\cdots+x_{n}^{\beta})}\left(\prod_{j=1}^{n}x_{j}\right)dx\leq
C\tau^{-2n/\beta}e^{-2\epsilon\tau}.\end{split}$

Applying (10), (11) to the right hand side of the equation

$\frac{1}{\tilde{c}_{0}(\tau)^{2}}-\frac{1}{{c}_{0}(\tau)^{2}}=\frac{e(\tau)}{\tilde{c}_{0}(\tau)^{4}(1+e(\tau)/\tilde{c}_{0}(\tau)^{2})},$

we can get the estimate in the lemma. ∎

##  5\. The $\hat{\mathcal{E}}$-condition

In this section, we introduce some classes of $C^{\infty}$ functions, which
are defined by the use of Newton polyhedra. When $F$ belongs to this class,
the behavior of $\tilde{\mathcal{B}}_{F}(\rho)$ is relatively easy to be
understood (see Theorem 3, below).

###  5.1. Newton polyhedra in the real case

Let $f$ be a real-valued $C^{\infty}$ function defined near the origin in
${\mathbb{R}}^{n}$. Let

(12)
$\sum_{\alpha\in{\mathbb{Z}}_{+}^{n}}c_{\alpha}x^{\alpha}=\sum_{\alpha\in{\mathbb{Z}}_{+}^{n}}c_{\alpha}x_{1}^{\alpha_{1}}\cdots
x_{n}^{\alpha_{n}}$

be the Taylor series of $f$ at the origin. The Newton polyhedron of $f$ is the
integral polyhedron

${\mathcal{N}}_{+}(f)=\mbox{the convex hull of the set
$\displaystyle\bigcup\\{\alpha+{\mathbb{R}}_{+}^{n}:\alpha\in S(f)$\\} in
${\mathbb{R}}_{+}^{n}$},$

where $S(f):=\\{\alpha\in{\mathbb{Z}}_{+}^{n}:c_{\alpha}\neq 0\\}$. In the
real case, we also say that $f$ is flat if ${\mathcal{N}}_{+}(f)=\emptyset$
and that $f$ is convenient if ${\mathcal{N}}_{+}(f)$ intersects all the axes.

###  5.2. The $\hat{\mathcal{E}}$-condition in the real case

Let $f$ be a $C^{\infty}$ function defined near the origin in
${\mathbb{R}}^{n}$. We say that $f$ admits the $\gamma$-part on an open
neighborhood $V$ of the origin in ${\mathbb{R}}^{n}$ if for any $x\in V$ the
limit

$\lim_{t\to 0}\frac{f(t^{a_{1}}x_{1},\ldots,t^{a_{n}}x_{n})}{t^{l}}$

exists for all pairs
$(a,l)=(a_{1},\ldots,a_{n},l)\in{\mathbb{Z}}_{+}^{n}\times{\mathbb{Z}}_{+}$
defining $\gamma$ (i.e., $\\{x\in P:\sum_{j=1}^{n}a_{j}x_{j}=l\\}=\gamma$). It
is known in [18] that when $f$ admits the $\gamma$-part, the above limits take
the same value for any $(a,l)$, which is denoted by $f_{\gamma}(x)$. We
consider $f_{\gamma}$ as the function on $V$, which is called the
$\gamma$-part of $f$. For a compact face $\gamma$ of ${\mathcal{N}}_{+}(f)$,
$f$ always admits the $\gamma$-part near the origin and $f_{\gamma}(x)$ equals
the polynomial
$\sum_{\alpha\in\gamma\cap{\mathbb{Z}}_{+}^{n}}c_{\alpha}x^{\alpha}$, where
$c_{\alpha}$ are as in (12).

######  Definition 1 ([18], [19]).

We say that $f$ satisfies the $\hat{\mathcal{E}}$-condition at the origin if
$f$ admits the $\gamma$-part for every proper face $\gamma$ of
${\mathcal{N}}_{+}(f)$.

######  Remark .

(1) If $f$ is real analytic near the origin or $f$ is convenient, then $f$
satisfies the $\hat{\mathcal{E}}$-condition (see [18], [19]). In particular,
in the one-dimensional case, nonflat functions satisfy the
$\hat{\mathcal{E}}$-condition.

(2) For example, $f(x_{1},x_{2})=x_{1}^{2}+e^{-1/x_{2}^{2}}$ does not satisfy
the $\hat{\mathcal{E}}$-condition.

Every $C^{\infty}$ function $f$ can be decomposed into an
$\hat{\mathcal{E}}$-function and a flat function.

######  Proposition 2.

For any $C^{\infty}$ function $f$ defined near the origin in
${\mathbb{R}}^{n}$, there exists a $C^{\infty}$ function $f_{0}$ satisfying
the $\hat{\mathcal{E}}$-condition at the origin such that $f-f_{0}$ is flat at
the origin.

###### Proof.

Since the proposition is obvious when $f$ does not vanish at the origin or $f$
is flat at the origin, we will only consider the other cases.

Let $p=(p_{1},\ldots,p_{n})\in{\mathbb{Z}}_{+}^{n}$ be a vertex of the Newton
polyhedron of $f$. We inductively define $C^{\infty}$ functions
$R_{0},R_{1},\ldots,R_{n}$ defined near the origin as follows.

Let $R_{0}(x)=f(x)$. Let $k$ is an integer with $1\leq k\leq n$. If $p_{k}\geq
1$, then there exists a $C^{\infty}$ function $R_{k}$ defined near the origin
such that

(13)
$R_{k-1}(x)=\sum_{j=0}^{p_{k}-1}c_{kj}(x_{1},\ldots,x_{k-1},x_{k+1},\ldots,x_{n})x_{k}^{j}+x_{k}^{p_{k}}R_{k}(x),$

where $c_{jk}$ are $C^{\infty}$ functions of $(n-1)$-variables. Note that (13)
is the Taylor expansion of $R_{k-1}$ with respect to the variable $x_{k}$. On
the other hand, if $p_{k}=0$, then set $R_{k}(x)=R_{k-1}(x)$.

By the use of the equations for $k=1,\ldots,n$, $f$ can be expressed as

(14) $f(x)=P(x)+x^{p}R_{n}(x),$

where $P$ is written in the form of the sum of a monomial times a $C^{\infty}$
function of $(n-1)$-variables, and $x^{p}=x_{1}^{p_{1}}\cdots x_{n}^{p_{n}}$.
It is easy to see that $R_{n}(0)=c_{p}$, where $c_{p}$ is as in (12) (i.e.,
$c_{p}$ is the coefficient of the term containing $x^{p}$ of the Taylor series
of $f$). Since $p$ is the vertex of the Newton polyhedron of $f$, $c_{p}$ does
not vanish, which implies $R_{n}(0)\neq 0$. It follows from Proposition 6.3 in
[18], we can see that $x^{p}R_{n}(x)$ satisfies the
$\hat{\mathcal{E}}$-condition.

Now, let us construct a $C^{\infty}$ function $f_{0}$ in the proposition by
induction on $n$. Note that every non-flat $C^{\infty}$ function of one
variable satisfies the $\hat{\mathcal{E}}$-condition. Let $f$ be a non-flat
$C^{\infty}$ function of $n$-variables with $f(0)=0$. Then $f$ can be
expressed as in (14) and $P$ takes the form of the sum of a monomial times a
$C^{\infty}$ function of $(n-1)$-variables. Now we assume that every
$C^{\infty}$ function of $(n-1)$-variables can be decomposed into an
$\hat{\mathcal{E}}$-function and a flat function. Then $P$ has the same type
decomposition and so a desirable $\hat{\mathcal{E}}$-function $f_{0}$ can be
obtained.

∎

###  5.3. The $\hat{\mathcal{E}}$-condition in the complex case

Now, let us consider a nonnegative $C^{\infty}$ function $F$ defined near the
origin in ${\mathbb{C}}^{n}$ with $F(0)=0$. If $F$ satisfies the condition
(C), then there exists a $C^{\infty}$ function $f$ defined near the origin in
${\mathbb{R}}^{n}$ such that
$f(|z_{1}|,\ldots,|z_{n}|)=F(z_{1},\ldots,z_{n})$. We say that $F$ satisfies
the $\hat{\mathcal{E}}$-condition at the origin, if $f$ satisfies the
$\hat{\mathcal{E}}$-condition at the origin. It is easy to see that the
$\hat{\mathcal{E}}$-condition of $F$ is independent of the choice of the
function $f$.

######  Proposition 3.

For any $C^{\infty}$ function $F$ defined near the origin in
${\mathbb{C}}^{n}$ satisfying the condition (C), there exists a $C^{\infty}$
function $F_{0}$ satisfying the condition (C) and the
$\hat{\mathcal{E}}$-condition at the origin such that $F-F_{0}$ is flat at the
origin.

######  Remark .

It is desirable to show that the additional conditions (A), (B) imply the
existence of $F_{0}$ such that $F-F_{0}$ is a nonnegative flat function. If
this implication can be shown, then the existence of $F_{0}$ can be removed in
the assumption in Theorem 2.

##  6\. Behavior of $\tilde{\mathcal{B}}_{F}(\rho)$ in the
$\hat{\mathcal{E}}$-case

In order to prove Theorem 2, it suffices to consider the case where $F$
satisfies the $\hat{\mathcal{E}}$-condition.

######  Theorem 3.

Suppose that $F$ is a $C^{\infty}$ function with $F(0)=0$ satisfying the
condition (C) in Section 2.2 and the $\hat{\mathcal{E}}$-condition at the
origin. If the principal face of ${\mathcal{N}}_{+}(F)$ is compact, then

$\tilde{\mathcal{B}}_{F}(\rho)=\frac{C(F_{*})}{\rho^{2+2/d_{F}}(\log\rho)^{m_{F}-1}}\cdot(1+o(\rho^{\varepsilon}))\quad\mbox{
as $\rho\to 0$,}$

where $d_{F}$ and $m_{F}$ are as in Section 2.1, $\varepsilon$ is a positive
constant, and $C(F_{*})$ is a positive constant depending only on the
principal part $F_{*}$ of $F$.

###### Proof.

Recall the expression of $\tilde{\mathcal{B}}_{F}(\rho)$:

$\begin{split}&\tilde{\mathcal{B}}_{F}(\rho)=\frac{1}{2\pi}\int_{0}^{\infty}e^{-2\rho\tau}\frac{\tau}{\tilde{c}_{0}(\tau)^{2}}d\tau\\\
&\mbox{ with }\,\,\tilde{c}_{0}(\tau)^{2}=\int_{{\mathbb{C}}^{n}}e^{-2\tau
F(z)}\Phi(z)dV(z).\end{split}$

This theorem can be directly shown by applying the following lemma to the
above integral formulas. ∎

######  Lemma 3.

Under the same assumptions on $F$ as those of Theorem 3,
$\tilde{c}_{0}(\tau)^{2}$ satisfies

(15)
$\tilde{c}_{0}(\tau)^{2}=c(F_{*})\tau^{-2/d_{F}}(\log\tau)^{m_{F}-1}\cdot(1+o(\tau^{-\varepsilon}))\quad\mbox{
as $\tau\to\infty$},$

where $\varepsilon$ is a positive constant and $c(F_{*})$ is a positive
constant depending only on the principal part $F_{*}$ of $F$.

The proof of the above lemma will be given in Section 8.

##  7\. Proof of Theorem 2

Let us give a proof of Theorem 2. We assume that $F$ is a $C^{\infty}$
function satisfying the conditions (A–D) in Section 2.2. Moreover, let $F_{0}$
be as in Theorem 2. It is easy to see that $F_{0}$ also satisfies the
conditions (B), (C).

For a positive number $M$, we define

$F_{1}(z)=F_{0}(z)+\sum_{j=1}^{n}|z_{j}|^{2M}.$

Since the principal face of ${\mathcal{N}}_{+}(F_{0})$ is compact, there
exists a positive constant $M$ such that the principal face of
${\mathcal{N}}_{+}(F_{1})$ is the same as that of ${\mathcal{N}}_{+}(F_{0})$.
Moreover, since $F_{1}$ is convenient, $F_{1}$ also satisfies the
$\hat{\mathcal{E}}$-condition at the origin (see Remark 2 (1) ). It is easy to
see that there exists a positive number $R$ such that

(16) $F_{0}(z)\leq F(z)\leq F_{1}(z)\quad\mbox{ for $z\in D(R)$.}$

For $j=0,1$, we define

$\tilde{\mathcal{B}}_{F_{j}}(\rho)=\frac{1}{2\pi}\int_{0}^{\infty}e^{-2\rho\tau}\frac{\tau}{g_{j}(\tau)}d\tau\quad\mbox{
for $\rho>0$,}$

where

$g_{j}(\tau)=\int_{{\mathbb{C}}^{n}}e^{-2\tau F_{j}(z)}\Phi(z)dV(z),$

where $\Phi$ is as in Section 4. From the relationship in (16), we see

(17)
$\tilde{\mathcal{B}}_{F_{0}}(\rho)\leq\tilde{\mathcal{B}}_{F}(\rho)\leq\tilde{\mathcal{B}}_{F_{1}}(\rho).$

Since it is easy to see that $F_{0},F_{1}$ satisfy the assumptions in Theorem
3, we have

(18)
$\tilde{\mathcal{B}}_{F_{j}}(\rho)=\frac{C(F_{*})}{\rho^{2+2/d_{F}}(\log\rho)^{m_{F}-1}}\cdot(1+o(\rho^{\varepsilon_{j}}))\quad\mbox{
as $\rho\to 0$},$

for $j=0,1$, where $\varepsilon_{0},\varepsilon_{1}$ are positive numbers.
Therefore, (17), (18) imply that

$\tilde{\mathcal{B}}_{F}(\rho)=\frac{C(F_{*})}{\rho^{2+2/d_{F}}(\log\rho)^{m_{F}-1}}\cdot(1+o(\rho^{\varepsilon}))\quad\mbox{
as $\rho\to 0$,}$

where $\varepsilon$ is the minimum of $\varepsilon_{1},\varepsilon_{2}$. From
the localization lemma in Proposition 1, we can obtain the equation (4) in the
theorem.

##  8\.  Asymptotic analysis of some Laplace integrals

The purpose of this section is to give a proof of Lemma 3. For this purpose,
we will investigate the behavior of a Lapace integral of the form

(19) $L_{f}(\tau)=\int_{{\mathbb{R}}_{+}^{n}}e^{-2\tau
f(x)}\varphi(x)\left(\prod_{j=1}^{n}x_{j}\right)dx,$

for large $\tau>0$, where $f$, $\varphi:V\to{\mathbb{R}}_{+}$ are nonnegative
and nonflat $C^{\infty}$ functions defined on a small open neighborhood $V$ of
the origin with $f(0)=0$ and $\varphi(0)=1$ and the support of $\varphi$ is
contained in $V$ of the origin. Since the compact support of $\varphi$ implies
the convergence of the above integral, $L_{f}$ can be considered as a
$C^{\infty}$ function defined on $(0,\infty)$.

The analysis in this section is based on the studies in [8], [18], [19], which
deal with oscillatory integrals and local zeta functions.

###  8.1. Newton data in the real case

Let $f$ be a nonflat real-valued $C^{\infty}$ function defined near the origin
in ${\mathbb{R}}^{n}$. Let $f$ admit the Taylor series
$\sum_{\alpha\in{\mathbb{Z}}_{+}^{n}}c_{\alpha}x^{\alpha}$ at the origin and
let ${\mathcal{N}}_{+}(f)$ be the Newton polyhedron of $f$ defined in Section
5.1. The Newton distance of $f$ is a nonnegative number

$d_{f}=\min\\{t\geq 0:(t,\ldots,t)\in{\mathcal{N}}_{+}(f)\\}.$

The minimum proper face of ${\mathcal{N}}_{+}(f)$ containing the point
$P_{f}:=(d_{f},\ldots,d_{f})$ is called the principal face of
${\mathcal{N}}_{+}(f)$, and is denoted by $\gamma_{*}$. The codimension of
$\gamma_{*}$ is called the Newton multiplicity, which is denoted by $m_{f}$.
When $\gamma_{*}$ is compact, the principal part of $f$ is defined by
$f_{*}(x)=\sum_{\alpha\in\gamma_{*}\cap{\mathbb{Z}}_{+}^{n}}c_{\alpha}x^{\alpha}.$

###  8.2. Meromorphic extension of some local zeta functions

Let us consider an integral of the form

(20)
$Z_{f}(s)=\int_{{\mathbb{R}}_{+}^{n}}f(x)^{s}\varphi(x)\left(\prod_{j=1}^{n}x_{j}\right)dx\quad\mbox{
for $s\in{\mathbb{C}}$},$

where $f,\varphi$ are the same as in (19). The convergence of the integral
easily implies that $Z_{f}$ can be regarded as a holomorphic function on the
right half-plane, which is called a local zeta function and is samely denoted
by $Z_{f}$.

The situation of analytic extension of the above local zeta function plays
crucial roles in the investigation of the behavior of Laplace integrals as
$\tau\to\infty$.

######  Theorem 4.

Suppose that

  1. (i)

$f$ satisfies the $\hat{\mathcal{E}}$-condition (see Section 5);

  2. (ii)

$f$ is Newton nondegenerate (see Section 9.3);

  3. (iii)

The principal face of ${\mathcal{N}}_{+}(f)$ is compact.

Then $Z_{f}$ can be analytically continued as a meromorphic function to the
whole complex plane, which will be samely denoted by $Z_{f}$. More precisely,
the set of the poles of $Z_{f}$ are contained in the set
$\\{-j/m:j\in{\mathbb{N}}\\}$ where $m$ is a positive integer.

Furthermore, the most right pole of $Z_{f}$ exists at $s=-2/d_{f}$ and its
order is $m_{f}$, where $d_{f},m_{f}$ are as in Section 8.1. The coefficient
of the leading term of the Laurent expansion of $Z_{f}$ at $s=-2/d_{f}$

$c_{{\tiny
Z}}(f_{*})=\lim_{s\to-2/d_{f}}\left(s+\frac{2}{d_{f}}\right)^{m_{f}}\cdot
Z_{f}(s)$

is a positive constant depending only on the principal part $f_{*}$ of $f$.

###### Proof.

The above theorem is a special case of Theorems 10.1 and 10.8 in [19]. Indeed,
$f$ satisfies the conditions (b) and (c) in Theorem 10.8.

∎

######  Remark .

The idea of the proof is based on the method of Varchenko in [23], [1] (see
also [8], [18], [19]). In his method, the toric resolution of singularities
based on the geometry of Newton polyhedra plays an essential role.

###  8.3. Asymptotic behavior of some Laplace integrals

From an information about the analytic continuation of $Z_{f}$ in Theorem 4,
we can see the behavior of the Laplace integral in (19) as $\tau\to\infty$.

######  Theorem 5.

Suppose that $f,\varphi$ satisfy the same conditions as in Theorem 4. Then
$L_{f}$ admits the asymptotic expansion: for any $N\in{\mathbb{N}}$, there
exists a positive constant $C_{N}$ such that

(21)
$\left|L_{f}(\tau)-\sum_{j=0}^{N}\sum_{k=1}^{n}c_{jk}\tau^{-j/m}(\log\tau)^{k-1}\right|<C_{N}\tau^{-N/m-\varepsilon},$

for $\tau\geq 2$, where $m$ is a positive integer determined by
${\mathcal{N}}_{+}(f)$, $\varepsilon$ is a positive number and $c_{jk}$ are
constants. In particular, the first term of the expansion can be expressed as

$L_{f}(\tau)=c_{L}(f_{*})\tau^{-2/d_{f}}(\log\tau)^{m_{f}-1}\cdot(1+o(\tau^{-\varepsilon}))\quad\mbox{
as $\tau\to\infty$},$

where $d_{f}$ and $m_{f}$ are as in Section 8.1 and $c_{L}(f_{*})$ is a
positive number depending only on the principal part $f_{*}$ of $f$.

###### Proof.

Define the fiber integral $H_{f}:{\mathbb{R}}_{+}\to{\mathbb{R}}$ as

$H_{f}(u)=\int_{W_{u}}\varphi(x)\left(\prod_{j=1}^{n}x_{j}\right)\omega,$

where $W_{u}:=\\{x\in{\mathbb{R}}^{n}:f(x)=u\\}$ and $\omega$ is the surface
element on $W_{u}$, which is determined by
$df\wedge\omega=dx_{1}\wedge\cdots\wedge dx_{n}$.

It is easy to see that the Laplace integral $L_{f}$ and the local zeta
function $Z_{f}$ can be represented by the use of $H_{f}$ as follows.

(22) $L_{f}(\tau)=\int_{0}^{\infty}e^{-2\tau u}H_{f}(u)du,$ (23)
$Z_{f}(s)=\int_{0}^{\infty}u^{s}H_{f}(u)du.$

Applying the inverse formula of the Mellin transform to (23), we have

$H_{f}(u)=\frac{1}{2\pi i}\int_{r-i\infty}^{r+i\infty}Z_{f}(s)u^{-s-1}ds,$

where $r>0$ and the integral contour follows the line ${\rm Re}(s)=r$ upwards.
The meromorphic extension of $Z_{f}$ in Theorem 4 implies that the deformation
of the integral contour as $r$ tends to $-\infty$ gives an asymptotic
expansion of $H_{f}(u)$ as $u\to 0$ by the residue formula. For any
$N\in{\mathbb{N}}$, there exists a positive constant $B_{N}$ such that

(24) $\left|H_{f}(u)-\sum_{j=0}^{N}\sum_{k=1}^{n}b_{jk}u^{j/m}(\log
u)^{k-1}\right|<B_{N}u^{N/m+\varepsilon},$

for $u\in(0,1/2)$, where $m$ is a positive integer determined by
${\mathcal{N}}_{+}(f)$, $\varepsilon$ is a positive number and $b_{jk}$ are
constants. We remark that the above deformation of the contour can be done. By
applying the asymptotic expansion (24) to (22), we can obtain (21).

In particular, we have

(25) $H_{f}(u)=c_{H}(f_{*})u^{2/d_{f}-1}(\log
u)^{m_{f}-1}\cdot(1+o(u^{\varepsilon}))\quad\mbox{ as $u\to 0$.}$

Substituting (25) into (22), we have

$L_{f}(\tau)=c_{L}(f_{*})\tau^{-2/d_{f}}(\log\tau)^{m_{f}-1}\cdot(1+o(\tau^{-\varepsilon}))\quad\mbox{
as $\tau\to\infty$.}$

∎

###  8.4. Proof of Lemma 3

By the use of the polar coordinates $z_{j}=x_{j}e^{i\theta_{j}}$, with
$x_{j}\geq 0,\theta_{j}\in{\mathbb{R}}$, for $j=1,\ldots,n$, then we have

$\begin{split}\tilde{c}_{0}(\tau)^{2}&=\int_{U}e^{-2\tau F(z)}\Phi(z)dV(z)\\\
&=(2\pi)^{n}\int_{{\mathbb{R}}_{+}^{n}}e^{-2\tau
f(x)}\left(\prod_{j=1}^{n}x_{j}\right)\varphi(x)dV(x),\end{split}$

where $\varphi$ is as in Section 4 and $f$ is a $C^{\infty}$ function defined
near the origin satisfying $f(|z_{1}|,\ldots,|z_{n}|)=F(z_{1},\ldots,z_{n})$.

From [16], the conditions (A-C) implies that $F_{\gamma}$ is positive on
$({\mathbb{R}}\setminus\\{0\\})^{n}$ for every compact face of
${\mathcal{N}}_{+}(F)$, which implies the Newton nondegeneracy of $f$ (see
Section 9.3). Moreover, it follows from the assumptions of $F$ that the the
principal face of $f$ is compact and $f$ satisfies the
$\hat{\mathcal{E}}$-condition. Therefore, since $f$ satisfies all the
assumptions in Theorem 5, we can obtain

$\tilde{c}_{0}(\tau)^{2}=C(f_{*})\tau^{-2/d_{f}}(\log\tau)^{m_{f}-1}\cdot(1+o(\tau^{-\varepsilon})),$

where $C(f_{*})$ is a positive constant depending only on $f_{*}$ and
$\varepsilon$ is a positive number. Since $d_{f}=d_{F}$ and $m_{f}=m_{F}$, we
can obtain (15) in the lemma.

##  9\. Appendix

###  9.1. Asymptotic expansion in (2)

Theorem 1 implies that the singularity of the Bergman kernel can be expressed
by the following complicated asymptotic expansion

$\Psi(\rho)\sim\sum_{j=0}^{\infty}\sum_{k=a_{j}}^{\infty}C_{jk}\frac{\rho^{j/m}}{(\log(1/\rho))^{k}}\quad\mbox{as
$\rho\to 0$,}$

where $m$ is a positive integer, $a_{j}$ are integers and $C_{jk}$ are real
numbers.

Let us explain the exact meaning of the above asymptotic expansion. For any
$N\in{\mathbb{N}}$, there exists a positive number $C_{N}$ such that

$\left|\Psi(\rho)-\sum_{j=0}^{N}C_{j}(\zeta)\rho^{j/m}\right|<C_{N}\rho^{N/m+\varepsilon}\quad\mbox{
for $\rho\in(0,\delta)$},$

where $\zeta=\log(1/\rho)^{-1}$ and every $C_{j}(\zeta)$ satisfies that, for
any $M\in{\mathbb{N}}$ satisfying $M\geq a_{j}$ there exists a positive number
$\tilde{C}_{M}$ such that

$\left|C_{j}(\zeta)-\sum_{k=a_{j}}^{M}C_{jk}\zeta^{k}\right|<\tilde{C}_{M}\zeta^{M+\tilde{\varepsilon}}\quad\mbox{
for $\zeta\in(0,\delta)$}.$

Here $\varepsilon$, $\tilde{\varepsilon}$, $\delta$ are some positive numbers.
Note that $\zeta\to 0$ if and only if $\rho\to 0$.

###  9.2. Convex geometry

Let us explain fundamental notions in the theory of convex polyhedra which are
necessary for our investigation. Refer to [24] for a general theory of convex
polyhedra.

For $(a,l)\in{\mathbb{R}}^{n}\times{\mathbb{R}}$, let $H(a,l)$ and
$H_{+}(a,l)$ be a hyperplane and a closed half-space in ${\mathbb{R}}^{n}$
defied by

$\begin{split}&H(a,l)=\\{x\in{\mathbb{R}}^{n};\langle a,x\rangle=l\\},\\\
&H_{+}(a,l)=\\{x\in{\mathbb{R}}^{n};\langle a,x\rangle\geq l\\},\end{split}$

respectively. Here $\langle a,x\rangle:=\sum_{j=1}^{n}a_{j}x_{j}$.

A (convex rational) polyhedron is an intersection of closed half-spaces: a set
$P\subset{\mathbb{R}}^{n}$ presented in the form
$P=\bigcap_{j=1}^{N}H_{+}(a^{j},l_{j})$ for some
$a^{1},\ldots,a^{N}\in{\mathbb{Z}}^{n}$ and
$l_{1}\ldots,l_{N}\in{\mathbb{Z}}$.

Let $P$ be a polyhedron in ${\mathbb{R}}^{n}$. A pair
$(a,l)\in{\mathbb{Z}}\times{\mathbb{Z}}$ is said to be valid for $P$ if $P$ is
contained in $H_{+}(a,l)$. A face of $P$ is any set of the form $F=P\cap
H(a,l)$, where $(a,l)$ is valid for $P$. Since $(0,0)$ is always valid, we
consider $P$ itself as a trivial face of $P$; the other faces are called
proper faces. Conversely, it is easy to see that any face is a polyhedron.
Considering the valid pair $(0,-1)$, we see that the empty set is always a
face of $P$. Indeed, $H_{+}(0,-1)={\mathbb{R}}^{n}$, but $H(0,-1)=\emptyset$.

The dimension of a face $F$ is the dimension of its affine hull (i.e., the
intersection of all affine flats that contain $F$), which is denoted by
$\dim(F)$. The faces of dimensions $0,1$ and $\dim(P)-1$ are called vertices,
edges and facets, respectively.

When $P$ is the Newton polyhedron of some non-flat $C^{\infty}$ function,
$\gamma$ is a compact face if and only if every valid pair
$(a,l)=(a_{1},\ldots,a_{n},l)$ defining $\gamma$ satisfies $a_{j}>0$ for any
$j$.

###  9.3. Newton nondegeneracy condition

We say that $f$ is Newton nondegenerate if the gradient of the $\gamma$-part
of $f$ has no zero in $({\mathbb{R}}\setminus\\{0\\})^{n}$ for every compact
face $\gamma$ of the Newton polyhedron ${\mathcal{N}}_{+}(f)$. This concept is
very important in the study of singularity theory.

When $\gamma$ is a compact face of ${\mathcal{N}}_{+}(f)$ with the valid pair
$(a_{1},\ldots,a_{n},l)$ defining $\gamma$, the following Euler identity
holds:

$lf_{\gamma}(x)=a_{1}x_{1}\frac{\partial f_{\gamma}}{\partial
x_{1}}(x)+\cdots+a_{n}x_{n}\frac{\partial f_{\gamma}}{\partial x_{n}}(x).$

It follows from this identity that if $f_{\gamma}$ has no zero in
$({\mathbb{R}}\setminus\\{0\\})^{n}$ for every compact face $\gamma$, then $f$
is Newton nondegenerate.

Acknowledgement The author greatly appreciates that the referee carefully read
this paper and gave many valuable comments. This work was supported by JSPS
KAKENHI Grant Numbers JP20K03656, JP20H00116

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# Solving order $3$ difference equations

Heba Bou KaedBey, Mark van Hoeij, and Man Cheung Tsui

###### Abstract.

We classify order $3$ linear difference operators over $\mathbb{C}(x)$ that
are solvable in terms of lower order difference operators. To prove this
result, we introduce the notions of restriction, induction, and absolute
irreducibility of difference modules, and classify modules that are
irreducible but not absolutely irreducible. We also show how restriction and
induction give coordinate-free formulations for sectioning and interlacing of
sequences.

###### Key words and phrases:

Difference Galois theory, difference algebra, linear recurrence equation

###### 2020 Mathematics Subject Classification:

12H10, 39A06

##  1\. Introduction

This paper classifies order $3$ linear difference operators of the form

$\phi^{3}+f\phi^{2}+g\phi+h,$

where $\phi(x)=x+1$ and $f,g,h\in\mathbb{C}(x)$, whose solutions are
expressible by the solutions of lower order operators. Previously [3]
determined the solutions of an arbitrary operator that are _Liouvillian
sequences_ , i.e., sequences generated by solutions of order $1$ operators
through repeated applications of difference ring operations, indefinite
summation, and interlacing of sequences. By defining _$2$ -expressible
sequences_ to be sequences generated in this way by solutions of order $2$
operators, we can state our main result as follows. (An algorithm implementing
this classification is given in the forthcoming paper [2].)

######  Theorem (Theorem 7.1 restated).

Let $L$ be an order $3$ difference operator over $\mathbb{C}(x)$. If $L$ is
_$2$ -solvable_, meaning it has a nonzero $2$-expressible solution, then at
least one of the following holds.

  1. (a)

$L$ admits a nontrivial factorization over $\mathbb{C}(x)$.

  2. (b)

$L$ is gauge equivalent to $\phi^{3}+f$ for some $f\in\mathbb{C}(x)$.

  3. (c)

$L$ is gauge equivalent to
$L_{2}^{\mathbin{\circledS}2}\mathbin{\circledS}L_{1}$ for some operators
$L_{1}$ and $L_{2}$ of orders $1$ and $2$, respectively, over $\mathbb{C}(x)$.

This theorem is the difference analogue of what M. Singer [18] proved for
differential operators. Aspects of [18] were subsequently generalized: M.
Singer [19] gave representation-theoretic criteria for solvability by lower
order operators; A. Person [13] used [19] to classify order $4$ operators that
are solvable in this way; M. van Hoeij [4] developed an algorithm for [18]; K.
A. Nguyen [11] and Nguyen-van der Put [12] generalized and streamlined [19],
and gave other solvability criteria. We take advantage of these developments
and reformulate results on difference operators in terms of difference modules
to simplify our proof of Case (c).

To analyze Cases (b) and (c), we introduce representation-theoretic ideas that
put the results of Hendriks-Singer [3] into a greater context. Viewing
difference modules as “twisted” $\mathbb{Z}$-representations allows us to
restrict and induce difference modules along subgroups of $\mathbb{Z}$. The
results of [3] are then seen as characterizing those difference modules that
are thus generated starting from one-dimensional modules. In a similar vein,
our Theorem 6.5 gives a criterion for when a difference module is induced from
lower-dimensional ones through the notion of absolute irreducibility.

Two natural operations on sequences—sectioning and interlacing—readily
correspond to restriction and induction. This was hinted at by the work of K.
Amano [1, Section 3.7], who also generalized [3] to Artinian simple module
algebras. By keeping to difference modules, we can show how restricted and
induced modules are coordinate-free formulations of sectioning and interlacing
of sequences in Section 8.

Our paper is organized as follows. Section 2 reviews basic facts about
difference operators and modules. Section 3 formally defines $2$-expressible
sequences and relates them to Eulerian groups. The symmetric power
construction as recalled in Section 4 gives us a criterion (Proposition 4.1)
to check for Case (c) of our main theorem. Section 5 defines induced and
restricted difference modules, while Section 6 proves various results about
them, most notably the characterization of absolute irreducibility in Theorem
6.5. Section 7 contains the proof of our main result, Theorem 7.1. Finally
Section 8 explicitly relates restricted and induced modules to sectioning and
interlacing of sequences.

Acknowledgement: The first and second authors were supported by NSF grant
CCF-2007959. We thank Ettore Aldrovandi for helpful conversations.

Assumptions: In this paper, $(F,\phi)$ is a _difference field_ , i.e., $F$ is
a field equipped with a field automorphism $\phi$; the subset
$C\coloneqq\left\\{f\in F\mid\phi(f)=f\right\\}$ is an algebraically closed
field of characteristic zero; $D\coloneqq F[\phi,\phi^{-1}]$ is the
noncommutative ring with multiplication $\phi\cdot f=\phi(f)\cdot\phi$.

##  2\. Background

In this section, we recall basic facts about difference modules and difference
Galois theory following [16, Chapter 1].

A _difference ring_ $(R,\phi)$ is a ring $R$ equipped with a ring automorphism
$\phi:R\to R$.

###  2.1. Difference modules

Unless otherwise specified, a _difference module_ or $D$-_module_ is a left
$D$-module of finite dimension as an $F$-vector space; the _dimension_ of the
$D$-module refers to this $F$-dimension. Let $M$ be a $D$-module. If
$\\{m_{1},\dots,m_{d}\\}$ is an $F$-basis of $M$ and
$\phi(m_{i})=\sum_{j}a_{ij}m_{j}$, then $M$ corresponds to the matrix
difference equation $\phi(Y)=(a_{ij})^{-1}Y$ according to [16, Section 1.4].

An element of $F[\phi]$ is a _difference operator_ over $(F,\phi)$. To a
nonzero difference operator $L$, we associate the (left) $D$-module

$M_{L}\coloneqq D/DL,$

and use this to define some notions for $L$. The _order_ of $L$ is defined as
the $F$-dimension of $M_{L}$. Two difference operators $L$ and $L^{\prime}$
are _gauge equivalent_ if $M_{L}$ and $M_{L^{\prime}}$ are isomorphic as
$D$-modules.

Let $M$ be a $D$-module. The _minimal operator_ of $m\in M$ is the unique
monic operator in $F[\phi]$ that generates the left ideal $\\{L\in D\mid
L(m)=0\\}$ of $D$. We call $m$ a _cyclic vector_ if $D\cdot m=M$, in which
case the minimal operator of $m$ has order $\dim_{F}M$. Therefore, a
difference operator is equivalent to a $D$-module with a choice of cyclic
vector. If the underlying field $F$ has an element $f$ for which
$\phi^{m}(f)\neq f$ for all $m\geq 1$, then every $D$-module has a cyclic
vector [3, Theorem B.2].

Let $R$ be a (left) $D$-algebra. The _solution space_ of an operator $L$ in
$R$ is the set $V_{R}(L)\coloneqq\\{f\in R\mid L(f)=0\\}$. The _solution
space_ of a $D$-module $M$ is the set

$V_{R}(M)\coloneqq\operatorname{Hom}_{D}(M,R).$

An element of $V_{R}(M)$ is called an $R$-_point_ of $M$. If $m$ is a cyclic
vector of $M$ and $L$ is the minimal operator of $m$, then the evaluation map

$V_{R}(M)\to V_{R}(L)\colon\quad s\mapsto s(m)$

is a bijection between the $R$-points of $M$ and the solutions of $L$ in $R$.
In general, $V_{R}(M)$ is a $C$-vector space of dimension $\leq\dim_{F}M$. We
say $V_{R}(M)$ is a _full solution space_ if it has $C$-dimension $\dim_{F}M$.

Let $L,L^{\prime}$ be operators. Then $\operatorname{GCRD}(L,L^{\prime})$ and
$\operatorname{LCLM}(L,L^{\prime})$ are the monic generators of the left ideal
$DL+DL^{\prime}$ and $DL\cap DL^{\prime}$ of $D$, respectively. Then
$V_{R}(\operatorname{LCLM}(L,L^{\prime}))=V_{R}(L)+V_{R}(L^{\prime})$.

###  2.2. Difference Galois theory

Let $M$ be a $D$-module with a corresponding matrix difference equation
$\phi(Y)=AY$, $A\in\operatorname{GL}_{n}(F)$. A _Picard-Vessiot ring_ for $M$
over $F$ is a difference ring extension $R/F$ such that:

  1. (1)

$\left\\{f\in R\mid\phi(f)=f\right\\}=C$;

  2. (2)

$R$ is a _simple difference ring_ , i.e., the only ideals $I$ of $R$ that
satisfy $\phi(I)\subseteq I$ are $0$ and $R$;

  3. (3)

$R=F[z_{ij},\det(Z)^{-1}\mid 1\leq i,j\leq n]$ for some
$Z=(z_{ij})\in\operatorname{GL}_{n}(R)$ satisfying $\phi(Z)=AZ$.

A Picard-Vessiot ring for $M$ exists, is unique up to an isomorphism of
$D$-algebras [16, Proposition 1.9], and is independent of the chosen matrix
difference equation [16, page 24].

Let $R$ be a Picard-Vessiot ring of $M$. The _difference Galois group_ of
$R/F$ is the group $\operatorname{Gal}(R/F)$ of $D$-algebra automorphisms of
$R$. Let $G=\operatorname{Gal}(R/F)$. By [16, Theorem 1.13], $G$ is a linear
algebraic group over $C$. For $\sigma\in G$ and $s\in V_{R}(M)$,
$(\sigma,s)\mapsto\sigma\circ s$ defines a $G$-action on $V_{R}(M)$ and
induces an injective homomorphism $G\to\operatorname{GL}(V_{R}(M))$ of
algebraic groups.

The total ring of fractions $K$ of a Picard-Vessiot ring of $M$ is called a
_total Picard-Vessiot ring_ of $M$. In place of $M$, we often refer to the
(total) Picard-Vessiot ring of an operator $L$ or matrix equation
corresponding to $M$. We also let $\operatorname{Gal}(K/F)$,
$\operatorname{Gal}(M)$, and $\operatorname{Gal}(L)$ refer to
$\operatorname{Gal}(R/F)$.

######  Proposition 2.1 ([3, Corollary A.7]).

Let $M$ be a $D$-module that has a cyclic vector and a total Picard-Vessiot
ring $K$. The map $N\mapsto V_{K}(N)$ defines a bijection between quotients of
$M$ and $\operatorname{Gal}(M)$-invariant subspaces of $V_{K}(M)$. Moreover,
$V_{K}(N)$ is a full solution space of $N$ for each quotient $N$ of $M$.

The Galois correspondence holds for difference modules.

######  Proposition 2.2.

Let $K/F$ be a total Picard-Vessiot extension with difference Galois group
$G$. Let $\mathcal{F}$ be the set of difference rings $E$ with $F\subseteq
E\subseteq K$ and such that every nonzero divisor of $E$ is a unit of $E$. Let
$\mathcal{G}$ be the set of algebraic subgroups of $G$.

  1. (a)

The maps $\mathcal{F}\to\mathcal{G}\colon E\mapsto\operatorname{Gal}(K/E)$ and
$\mathcal{G}\to\mathcal{F}\colon H\mapsto K^{H}$ are inverses;

  2. (b)

If $E/F$ is an intermediate total Picard-Vessiot extension of $K/F$ then
$\operatorname{Gal}(K/E)\trianglelefteq G$.

###### Proof.

(a) is [16, Theorem 1.29], and (b) follows from the computation

$\sigma\operatorname{Gal}(K/E)\sigma^{-1}=\operatorname{Gal}(K/\sigma(E))=\operatorname{Gal}(K/E)$

which holds for all $\sigma\in G$ since $E/F$ is a total Picard-Vessiot
extension. ∎

##  3\. 2-expressible sequences

We first recall basic constructions on sequences (indefinite summation,
interlacing, Liouvillian sequences) as in [3, Section 3]. We then define
$2$-expressible sequences and relate them to Eulerian groups in Theorem 3.2.

Let $\mathbb{N}=\\{0,1,2,\dots\\}$. We define $\mathcal{S}$ to be the quotient
set of functions $s:\mathbb{N}\to C$ under the equivalence relation $s\sim t$
if $s(n)=t(n)$ for all but finitely many $n$. An element of $\mathcal{S}$ is
called a _sequence_ and is represented as a function $s:\mathbb{N}\to C$ or as
a list $(s(0),s(1),s(2),\dots)$. The set $\mathcal{S}$ becomes a difference
ring under function addition and multiplication, and $\phi(s(n))\coloneqq
s(n+1)$. Since rational functions have finitely many poles, $C(x)$ embeds into
$\mathcal{S}$ by evaluation at $\mathbb{N}$. We work with $\mathcal{S}$ since
any difference operator over $C(x)$ has a full solution space in $\mathcal{S}$
by [14, Theorem 8.2.1] (see also [16, Proposition 4.1]).

We construct new sequences from old ones as follows. For $s\in\mathcal{S}$,

$\Sigma s\coloneqq\\{t\in\mathcal{S}\mid(\phi-1)t=s\\}$

is the set of _indefinite sums_ of $s$. The _interlacing_ of
$s_{0},s_{1},\dots,s_{m-1}\in\mathcal{S}$ is the sequence $t$ defined by
$t(mn+i)=s_{i}(n)$ for $n\geq 0$ and $0\leq i<m$. The $m$-_interlacing_ of $s$
_with zeros_ is defined as the interlacing of $s$ with $(m-1)$-many zero
sequences. Thus the interlacing of $s_{0},s_{1},\dots,s_{m-1}\in\mathcal{S}$
is the sum $t_{0}+\phi(t_{1})+\dots+\phi^{m-1}(t_{m-1})$ where $t_{i}$ is the
$m$-interlacing of $s_{i}$ with zeros.

Recall that the ring of _Liouvillian sequences_ $\mathcal{L}$ is defined as
the smallest _difference subring_ 111in other words, closed under
$+,\cdot,\phi,\phi^{-1}$ $R$ of $\mathcal{S}$ that satisfy the following
conditions.

  1. (1)

Any order $1$ operator over $C(x)$ has a full solution space in $R$.

  2. (2)

If $s\in R$, then $\Sigma s\subseteq R$.

  3. (3)

If $s\in R$, then the $m$-interlacing of $s$ with zeros is in $R$ for all
$m\geq 1$.

Condition (3) and the closure of $\mathcal{L}$ under difference ring
operations imply that $\mathcal{L}$ is closed under arbitrary interlacing.

We similarly define what it means to generate sequences from solutions of
order $2$ operators analogous to the Eulerian solutions considered in [18].

######  Definition 3.1.

We define $\mathcal{E}$ to be the smallest difference subring $R$ of
$\mathcal{S}$ that satisfy conditions (2) and (3) above, as well as

  1. (4)

Any order $2$ operator over $C(x)$ has a full solution space in $R$.

An element of $\mathcal{E}$ is called _$2$ -expressible_. An
$\mathcal{S}$-point $s\colon M\to\mathcal{S}$ of a $D$-module $M$ is _$2$
-expressible_ if its image is contained in $\mathcal{E}$.

Clearly a solution of an operator $L$ is $2$-expressible if and only if the
corresponding $\mathcal{S}$-point of $M_{L}$ is $2$-expressible.

[3, Theorem 3.4] shows how Liouvillian sequences correspond to solvable Galois
groups. We now give an analogue for $2$-expressible sequences and Eulerian
Galois groups. Recall from [18, page 667] that an algebraic group $G$ is said
to be _Eulerian_ if there exist subgroups $1=G_{0}\subset\dots\subset G_{n}=G$
such that $G_{i}\triangleleft G_{i+1}$ and $G_{i+1}/G_{i}$ is finite or
isomorphic to $\mathbb{G}_{a}$, $\mathbb{G}_{m}$, or
$\operatorname{PSL}_{2}(C)$. Eulerian groups are closed under taking closed
subgroup, quotients, and direct products [18, Lemma 2.2]. Since the connected
closed subgroups of $\operatorname{SL}_{2}(C)$ are either solvable or
$\operatorname{SL}_{2}(C)$ by [6, page 31] (see also [7, page 7] and [15, page
28]), the Galois group of an order $2$ operator is Eulerian.

######  Theorem 3.2.

Let $s\in\mathcal{E}$. Then there is a nonzero operator $L$ with
$\operatorname{Gal}(L)$ Eulerian and $s\in
V_{\mathcal{S}}(L)\subseteq\mathcal{E}$.

###### Proof.

We must show the following sets $A$ and $B$ contain $\mathcal{E}$.

$\displaystyle A$ $\displaystyle=\\{s\in\mathcal{S}\mid\exists\,L_{1}\neq
0\colon L_{1}(s)=0,\operatorname{Gal}(L_{1})\text{ is Eulerian}\\}$
$\displaystyle B$ $\displaystyle=\\{s\in\mathcal{S}\mid\exists\,L_{2}\neq
0\colon s\in V_{\mathcal{S}}(L_{2})\subseteq\mathcal{E}\\}$

Then $s\in\mathcal{E}$ implies $s\in A$ and $s\in B$, providing operators
$L_{1}$ and $L_{2}$, and we can take $L$ to be their greatest common right
divisor.

It suffices to show that $A$ and $B$ are difference rings closed under
conditions (2)–(4), since, by definition, $\mathcal{E}$ is the intersection of
such rings. The solutions of order $2$ operators over $C(x)$ are clearly in
$B$, and in $A$ by the above discussion.

Let $s,t\in A$ be solutions to operators with Eulerian Galois groups $G,H$,
resp. The proof of [3, Theorem 3.4] shows that $\phi(s)$, $\Sigma s$, and the
$m$-interlacing of $s$ with zeros are solutions to operators whose Galois
groups are $G$, a subgroup of the semidirect product of $G$ and
$\mathbb{G}_{a}^{n}$ for some $n$, or a finite cyclic extension of $G^{m}$.
Moreover, $st$ and $s-t$ lie in a composite Picard-Vessiot ring with Galois
group $G^{\prime}\leq G\times H$. By [3, Lemma A.8], both $st$ and $s-t$ are
solutions to operators with Galois groups that are quotients of $G^{\prime}$.
Since Eulerian groups are closed under taking closed subgroup, quotients, and
direct products [18, Lemma 2.2], $\phi(s),\Sigma s$, the $m$-interlacing of
$s$, $st$, and $s-t$ satisfy operators with Eulerian Galois groups and so are
in $A$. Thus $\mathcal{E}\subseteq A$.

Let $s,t\in B$ be solutions to operators $L,R$ with
$V_{\mathcal{S}}(L),V_{\mathcal{S}}(R)\subseteq\mathcal{E}$. Then

$\displaystyle s+t$ $\displaystyle\in
V_{\mathcal{S}}(\operatorname{LCLM}(L,R))=V_{\mathcal{S}}(L)+V_{\mathcal{S}}(R)\subseteq\mathcal{E},$
$\displaystyle st$ $\displaystyle\in
V_{\mathcal{S}}(L\mathbin{\circledS}R)\subseteq\mathcal{E},$
$\displaystyle\phi^{i}(s)$ $\displaystyle\in V_{\mathcal{S}}(\phi^{i}\circ
L\circ\phi^{-i})=\phi^{i}(V_{\mathcal{S}}(L))\subseteq\mathcal{E}\text{ for
all }i.$

Here $V(L\mathbin{\circledS}R)\subseteq\mathcal{E}$ uses equation (1) in
Section 4.

Finally if $s\in B$ and $L$ is the minimal operator of $s$, say of order $n$,
then replacing $x$ by $x/m$ and $\phi$ by $\phi^{m}$ gives an order $mn$
operator $L^{\prime}$ for the $m$-interlacing of $s$ with zeros. By Lemma 3.4,
$V_{\mathcal{S}}(L\mid_{x\mapsto x+i/m})\subseteq\mathcal{E}$. Appropriate
$\phi$-shifts of the $m$-interlacing of these elements with zeros span
$V_{\mathcal{S}}(L^{\prime})$. Thus
$V_{\mathcal{S}}(L^{\prime})\subseteq\mathcal{E}$ showing that $B$ is closed
under $m$-interlacing. As before, the closure properties imply
$\mathcal{E}\subseteq B$. ∎

######  Definition 3.3.

$L$ is _$2$ -solvable_ if it has a nonzero 2-expressible solution, and it is
_fully $2$-solvable_ if it has a basis of 2-expressible solutions.

######  Lemma 3.4.

Let $L$ be a difference operator over $C(x)$, and let $c\in C$. If $L$ is
fully $2$-solvable then so is $L\mid_{x\mapsto x+c}$.

###### Proof.

Proof by induction. The base case (order 2) is trivial. Difference ring
operations and the indefinite sum case follow directly by induction. For the
$m$-interlacing case, use induction with $c$ replaced with $c/m$. ∎

######  Corollary 3.5.

$L$ is $2$-solvable if and only if $L$ has a fully 2-solvable right factor.

###### Proof.

If $s\in\mathcal{E}$ is a nonzero solution of $L$, and if $R$ is its minimal
operator, then $R$ is a right factor and Theorem 3.2 says that it is fully
$2$-solvable. ∎

A question remains: is every operator with an Eulerian Galois group
$2$-solvable? For order $3$ this follows from the proof in Section 7.

##  4\. Symmetric powers

We recall some linear algebraic operations (tensor product, symmetric power)
for combining $D$-modules and prove a preliminary structure theorem
(Proposition 4.1).

Let $M$ and $N$ be $D$-modules. The tensor product $M\otimes_{F}N$ is a
$D$-module under the diagonal action $\phi(m\otimes
n)\coloneqq\phi(m)\otimes\phi(n)$. The tensor power $M^{\otimes d}$ is
naturally a $D$-module, as is the subset $S^{d}(M)\subseteq M^{\otimes d}$ of
elements fixed under permutations of tensor factors. We call $S^{d}(M)$ the
$d$th _symmetric power_ of $M$.

Let $L_{1},L_{2}\in D$. The _symmetric product_ of $L_{1}$ and $L_{2}$,
denoted by $L_{1}\mathbin{\circledS}L_{2}$, is the minimal operator of
$1\otimes 1$ in $M_{L_{1}}\otimes_{F}M_{L_{2}}$. As with [16, Corollary 2.19],
one has

$V_{R}(L_{1}\mathbin{\circledS}L_{2})=\operatorname{span}_{C}\left\\{u\cdot
v\mid u\in V_{R}(L_{1}),v\in V_{R}(L_{2})\right\\}$ (1)

for any $D$-algebra $R$. We let
$L^{\mathbin{\circledS}2}=L\mathbin{\circledS}L$ and note that $D(1\otimes
1)\subseteq S^{2}(M_{L})$.

Here is a criterion on when a $D$-module of dimension $3$ is an almost
symmetric square. This is an analogue of a result by Fano, Singer, van
Hoeij–van der Put [5, Proposition 3.1], and Nguyen–van der Put [12, Corollary
2.2]. We follow the latter presentation.

######  Proposition 4.1.

Let $F$ be a $C_{1}$-field. Let $M$ be an irreducible $D$-module of dimension
$3$. Then $S^{2}(M)$ has a $D$-submodule of dimension $1$ if and only if
$M\cong S^{2}(N)\otimes P$ for some $D$-modules $N$ and $P$ of dimensions $2$
and $1$, respectively.

###### Proof.

$\Leftarrow$: The canonical $D$-module epimorphism
$S^{2}(S^{2}(N))\twoheadrightarrow S^{4}(N)$ has a kernel of dimension $1$.
Hence $S^{2}(M)\cong S^{2}(S^{2}(N))\otimes S^{2}(P)$ has a submodule of
dimension $1$ as well.

$\Rightarrow$: Let $q\in S^{2}(M)$ span a $D$-submodule of dimension $1$. View
$q$ as its associated symmetric bilinear form on $M^{*}$. Then $\\{x\in
M^{*}\mid q(x,M^{*})=0\\}$ is a $D$-module since $\phi(q)=rq$ for some $r\in
F^{*}$. This submodule is zero since $M^{*}$ is irreducible. So $q$ is
nondegenerate.

A sequence of elementary linear transformations rewrites $q$ to an
$F$-multiple of $x_{2}^{2}-x_{1}x_{3}$ in some $F$-basis $x_{1},x_{2},x_{3}$
of $M$. Alternatively, Witt’s decomposition theorem [8, page 12] makes $q$ an
$F$-multiple of the sum of an anisotropic form $x_{2}^{2}$ and a hyperbolic
form $-x_{1}x_{3}$.

Write $\phi(x_{i})=\sum_{j}g_{ij}x_{j}$ with
$(g_{ij})\in\operatorname{GL}_{3}(F)$. Since $q$ spans a 1-dimensional
submodule, we have $\phi(x_{2}^{2}-x_{1}x_{3})=r(x_{2}^{2}-x_{1}x_{3})$ for
some $r\in F^{*}$. Matching $x_{i}x_{j}$-coefficients gives the following
relations.

$\displaystyle[x_{1}^{2}]$ $\displaystyle:$ $\displaystyle 0$
$\displaystyle=g_{21}^{2}-g_{11}g_{31}$ (2) $\displaystyle[x_{3}^{2}]$
$\displaystyle:$ $\displaystyle 0$ $\displaystyle=g_{23}^{2}-g_{13}g_{33}$
$\displaystyle[x_{1}x_{2}]$ $\displaystyle:$ $\displaystyle 0$
$\displaystyle=2g_{21}g_{22}-g_{11}g_{32}-g_{31}g_{12}$
$\displaystyle[x_{2}x_{3}]$ $\displaystyle:$ $\displaystyle 0$
$\displaystyle=2g_{22}g_{23}-g_{12}g_{33}-g_{32}g_{13}$
$\displaystyle[x_{2}^{2}]+[x_{1}x_{3}]$ $\displaystyle:$ $\displaystyle 0$
$\displaystyle=(g_{22}^{2}-g_{12}g_{32})+(2g_{21}g_{23}-g_{11}g_{33}-g_{31}g_{13})$

Let $N$ and $P$ be $F$-vector spaces with respective bases $\\{n_{1},n_{2}\\}$
and $\\{p\\}$. The $F$-linear map $S^{2}(N)\otimes P\to M$ taking
$n_{i}n_{j}\otimes p$ to $m_{ij}$ will be a $D$-module isomorphism if we can
find $a_{ij},b\in F$ defining actions $\phi(n_{i})=\sum a_{ij}n_{j}$ on $N$
and $\phi(p)=b\cdot p$ on $P$ that satisfy

$\begin{bmatrix}g_{11}&g_{12}&g_{13}\\\ g_{21}&g_{22}&g_{23}\\\
g_{31}&g_{32}&g_{33}\\\
\end{bmatrix}=b\begin{bmatrix}a_{11}^{2}&2a_{11}a_{12}&a_{12}^{2}\\\
a_{11}a_{21}&a_{11}a_{22}+a_{12}a_{21}&a_{12}a_{22}\\\
a_{21}^{2}&2a_{21}a_{22}&a_{22}^{2}\\\ \end{bmatrix}.$ (3)

From (2) and $\det(g_{ij})\neq 0$, either $g_{11}g_{21}\neq 0$,
$g_{21}g_{31}\neq 0$, $g_{13}g_{23}\neq 0$, $g_{23}g_{33}\neq 0$, or
$(g_{ij})$ is diagonal or skew-diagonal. Reflecting the square matrices
$(a_{ij})$ and $(g_{ij})$ preserve the form of (2) and (3), so we may assume
$(g_{ij})$ is diagonal or $g_{11}g_{21}\neq 0$. In the former case, set
$b=g_{11}$, $a_{11}=1$, $a_{22}=g_{22}/g_{11}$, and $a_{12}=a_{21}=0$. In the
latter case, set $b=g_{11}$, $a_{11}=1$, $a_{12}=g_{12}/2g_{11}$,
$a_{21}=g_{21}/g_{11}$, and $a_{22}=g_{32}/2g_{21}$; the relations (3) follow
from (2) by a Gröbner basis computation in Appendix A. ∎

##  5\. Restriction and induction

We now define representation-theoretic operations on difference modules.

######  Definition 5.1.

Let $D_{t}=F[\phi^{t},\phi^{-t}]$.

  1. (1)

The _restriction_ of a $D$-module $M$ to $D_{t}$ is the $D_{t}$-module $M$ and
denoted by $M\mathord{\downarrow}^{1}_{t}$.

  2. (2)

The _induction_ of a $D_{t}$-module $N$ to $D$ is the $D$-module
$D\otimes_{D_{t}}N$ and denoted by $N\mathord{\uparrow}^{1}_{t}$.

  3. (3)

The _restriction_ $L\mathord{\downarrow}^{1}_{t}$ of an operator $L\in D$ is
the minimal operator of $1\in M_{L}\mathord{\downarrow}^{1}_{t}$. In other
words, $L\mathord{\downarrow}^{1}_{t}$ is the monic element of minimal order
in $D_{t}\cap DL$.

Let $M$ be a $D$-module with basis $\mathcal{B}$ and cyclic vector $m$, and
with corresponding matrix equation $\phi(Y)=AY$ and operator $L$. Then
$M\mathord{\downarrow}^{1}_{t}$ has basis $\mathcal{B}$ with corresponding
matrix equation $\phi^{t}(Y)=A_{t}Y$, where

$A_{t}\coloneqq\phi^{t-1}(A)\cdots\phi^{2}(A)\phi(A)A.$

Since $m\in M\mathord{\downarrow}^{1}_{t}$ might not be cyclic, the operator
$L\mathord{\downarrow}^{1}_{t}$ only corresponds to a $D_{t}$-submodule of
$M\mathord{\downarrow}^{1}_{t}$.

Let $N$ be a $D_{t}$-module with basis $\mathcal{C}$ and cyclic vector $n$,
and with corresponding matrix equation $\phi^{t}(Y)=BY$ and operator $R\in
D_{t}$. Then $N\mathord{\uparrow}^{1}_{t}$ has basis
$\bigcup_{i=0}^{t-1}\phi^{i}\otimes\mathcal{C}$ and cyclic vector $n$, and
corresponding matrix equation

$\phi(Y)=\tilde{B}Y,\qquad\tilde{B}\coloneqq\begin{bmatrix}0&I_{(t-1)d}\\\
B&0\end{bmatrix}$

where $d=\dim_{F}N$. The minimal operator of $n$ is the same $R$, now viewed
as an element of $D\supseteq D_{t}$.

The table below summarizes the relationship we just discussed about difference
modules, matrix difference equations, and difference operators.

Module | Matrix Equation | Operator
---|---|---
$M$ | $\phi(Y)=AY$ | $L\in D$
$M\mathord{\downarrow}^{1}_{t}$ | $\phi^{t}(Y)=A_{t}Y$ |  $L\mathord{\downarrow}^{1}_{t}\in D_{t}$ if $m\in M\mathord{\downarrow}^{1}_{t}$ is cyclic
$N$ | $\phi^{t}(Y)=BY$ | $R\in D_{t}$
$N\mathord{\uparrow}^{1}_{t}$ | $\phi(Y)=\tilde{B}Y$ | $R\in D$

Many facts translate well into this language:

######  Proposition 5.2.

Let $(F,\phi)=(C(x),x\mapsto x+1)$ and $M$ be a $D$-module.

  1. (1)

If $\dim_{F}M>1$ and $\operatorname{Gal}(M)$ is solvable, then
$M\mathord{\downarrow}^{1}_{t}$ is reducible for some $t\geq 1$. (I.e. $M$ is
not absolutely irreducible; see Definition 6.4.)

  2. (2)

For some $t\geq 1$, the total Picard-Vessiot ring of
$M\mathord{\downarrow}^{1}_{t}$ is a difference field $(K,\phi^{t})$ and
$\operatorname{Gal}(M\mathord{\downarrow}^{1}_{t})=\operatorname{Gal}(M)^{\circ}$.

###### Proof.

(1): We are done if $M$ is reducible. Suppose $M$ is irreducible with an
associated operator $L$. Since $\operatorname{Gal}(L)$ is solvable, some
nonzero solution of $L$ is a Liouvillian sequence [3, Theorem 3.4]. By [3,
Theorem 5.1], some nonzero solution of $L$ is an interlacing of hypergeometric
sequences. By [3, Corollary 4.3], $L$ is gauge equivalent to some $\phi^{t}+f$
for some $f\in F$. Thus $D_{t}/D_{t}(\phi^{t}+f)$ is a $1$-dimensional
quotient of $(D/DL)\mathord{\downarrow}^{1}_{t}\cong
M\mathord{\downarrow}^{1}_{t}$, and $M\mathord{\downarrow}^{1}_{t}$ is
reducible.

(2): The Picard-Vessiot ring $S$ of $M$ has the form
$\bigoplus_{i=0}^{t-1}\phi^{i}(R)$ for some difference ring $(R,\phi^{t})$
where $R$ is an integral domain. By [16, Lemma 1.26],
$M\mathord{\downarrow}_{t}^{1}$ has the Picard-Vessiot ring $(R,\phi^{t})$ so
we may set $K\coloneqq\operatorname{Frac}(R)$.

By [16, Proposition 1.20], $S$ (resp. $R$) is isomorphic to the coordinate
ring of the algebraic group representing $\operatorname{Gal}(M)$ (resp.
$\operatorname{Gal}(M\mathord{\downarrow}_{t}^{1})$) after base change to $F$.
Since $R$ is a connected component of $S$, we have
$\operatorname{Gal}(M\mathord{\downarrow}_{t}^{1})=\operatorname{Gal}(M)^{\circ}$.
∎

##  6\. Semisimplicity and absolute irreducibility

We further decompose irreducible modules by representation theory, culminating
in Theorem 6.5 on restriction-reducibility. For this, we also need semisimple
modules; see [9, Chapter 17] for general facts.

Like differential modules [15, Exercise 2.38 (4)], the tensor product of two
semisimple $D$-modules is again semisimple. Therefore the class of semisimple
$D$-modules is closed under linear algebraic operations like taking
submodules, quotients, tensor products, symmetric powers, and direct sums. We
now show that restriction and induction also preserve semisimplicity.

######  Proposition 6.1.

Let $M$ be an irreducible $D$-module and $N$ an irreducible $D_{t}$-module.
Then $M\mathord{\downarrow}^{1}_{t}$ and $N\mathord{\uparrow}^{1}_{t}$ are
semisimple.

###### Proof.

The proof is standard. To show $M\mathord{\downarrow}^{1}_{t}$ is semisimple,
let $P$ be an irreducible $D_{t}$-submodule of $M$. Then the $D$-submodule
$P+\phi(P)+\dots+\phi^{t-1}(P)$ equals $M$ since $M$ is irreducible. Thus
$M\mathord{\downarrow}^{1}_{t}$ is a sum of the irreducible $D_{t}$-submodules
$\phi^{i}(P)$.

We next show $N\mathord{\uparrow}^{1}_{t}$ is semisimple. Let $Q$ be a
$D$-submodule of $N\mathord{\uparrow}^{1}_{t}$. Since
$N\mathord{\uparrow}^{1}_{t}\mathord{\downarrow}^{1}_{t}=\bigoplus_{i}\phi^{i}\otimes
N$ is semisimple, there exists a projection $\alpha\colon
N\mathord{\uparrow}^{1}_{t}\mathord{\downarrow}^{1}_{t}\twoheadrightarrow
Q\mathord{\downarrow}^{1}_{t}$ of $D_{t}$-modules. The map

$\tilde{\alpha}\colon N\mathord{\uparrow}^{1}_{t}\twoheadrightarrow
N\mathord{\uparrow}^{1}_{t}:\quad
n\mapsto\frac{1}{t}\sum_{i=0}^{t-1}\left(\phi^{-i}\circ\alpha\circ\phi^{i}\right)(n)$

is now a $D$-module homomorphism. Moreover $\tilde{\alpha}$ is a projection
with image $Q$. This gives the $D$-module decomposition

$N\mathord{\uparrow}^{1}_{t}=\operatorname{im}(\tilde{\alpha})\oplus\ker(\tilde{\alpha})=Q\oplus\ker(\tilde{\alpha}).$

∎

The above is the weak analogue of Clifford’s theorem [10, Theorem 23.1]. Here
is part of the strong analogue [10, Theorem 23.3]:

######  Proposition 6.2.

Let $M$ be an irreducible $D$-module. Let
$M\mathord{\downarrow}^{1}_{t}=M_{1}^{n_{1}}\oplus\dots\oplus M_{r}^{n_{r}}$
be the decomposition of $M$ into non-isomorphic, nonzero $D_{t}$-modules
$M_{i}$. Then $\phi$ permutes the $M_{i}^{n_{i}}$ transitively. In particular,
we have $n_{1}=\cdots=n_{r}$ and $\dim_{F}M_{1}=\cdots=\dim_{F}M_{r}$.

###### Proof.

The group $\langle\phi\rangle$ permutes the $M_{i}^{n_{i}}$ by the uniqueness
of the isotypic decomposition. This action is transitive because $M$ is
irreducible. Therefore $\dim_{F}M_{i}^{n_{i}}=n_{i}\dim_{F}M_{i}$ are all
equal. Finally since $\phi(M_{i}^{n_{i}})=(\phi M_{i})^{n_{i}}$,
$\langle\phi\rangle$ induces a (transitive) permutation on the isomorphism
classes of the $M_{i}$, and so $\dim_{F}M_{i}$ are all equal. ∎

In fact the $n_{i}$ are 1, see also Corollary 6.8 below. Here is an analogue
of [17, Section 7.4].

######  Theorem 6.3 (Mackey’s irreducibility criterion).

Let $N$ be an irreducible $D_{t}$-module. Then $N\mathord{\uparrow}^{1}_{t}$
is irreducible if and only if $\phi^{i}\otimes N$ is not isomorphic to $N$ as
$D_{t}$-modules for each $i=1,2,\dots,t-1$.

###### Proof.

Tensor-hom adjunction (or Frobenius reciprocity) gives

$\begin{split}\operatorname{End}_{D}(N\mathord{\uparrow}^{1}_{t})&\cong\operatorname{Hom}_{D_{t}}(N,N\mathord{\uparrow}^{1}_{t}\mathord{\downarrow}^{1}_{t})=\bigoplus_{i=0}^{t-1}\operatorname{Hom}_{D_{t}}(N,\phi^{i}\otimes
N).\end{split}$

By Proposition 6.1, $N\mathord{\uparrow}^{1}_{t}$ is semisimple. Thus
$N\mathord{\uparrow}^{1}_{t}$ is irreducible precisely when
$\operatorname{End}_{D_{t}}(N\mathord{\uparrow}^{1}_{t})=C$, i.e., when
$\operatorname{Hom}_{D_{t}}(N,\phi^{i}\otimes N)=0$ for $i=1,2,\dots,t-1$ by
the above computation. Since the nonzero elements of
$\operatorname{Hom}_{D_{t}}(N,\phi^{i}\otimes N)$ are isomorphisms,
$\operatorname{Hom}_{D_{t}}(N,\phi^{i}\otimes N)=0$ holds if and only if
$N\not\cong\phi^{i}\otimes N$. ∎

This characterizes the irreducibility of induced modules. We now do the same
for restricted modules.

######  Definition 6.4.

A $D$-module $M$ is _absolutely irreducible_ if
$M\mathord{\downarrow}^{1}_{t}$ is irreducible for all $t\geq 1$. An operator
$L$ is _absolutely irreducible_ if $M_{L}$ is absolutely irreducible. We say
that $L$ or $M$ is _restriction-reducible_ if it is irreducible but not
absolutely irreducible.

######  Theorem 6.5.

If a $D$-module $M$ is restriction-reducible with
$M\mathord{\downarrow}^{1}_{t}$ reducible, then $M\cong
N\mathord{\uparrow}^{1}_{s}$ for some divisor $s>1$ of $t$ and some
$D_{s}$-module $N$.

###### Proof.

Consider a minimal divisor $s>1$ of $t$ for which
$M\mathord{\downarrow}^{1}_{s}$ is reducible and let $N\subsetneq
M\mathord{\downarrow}^{1}_{s}$ be an irreducible $D_{s}$-submodule. Since $M$
is irreducible, the map

$N\mathord{\uparrow}^{1}_{s}\to M\colon\quad\phi^{i}\otimes
n\mapsto\phi^{i}(n)$

is surjective. To finish the proof, it suffices to show that
$N\mathord{\uparrow}^{1}_{s}$ is irreducible, for then the above map is an
isomorphism.

Suppose $N\mathord{\uparrow}^{1}_{s}$ is reducible. By Theorem 6.3, we have
$N\cong\phi^{i}(N)$ as $D_{s}$-modules for some $0<i<t$. Using $\phi^{s}(N)=N$
and iterating the isomorphism $N\cong\phi^{i}(N)\cong\phi^{2i}(N)\cong\cdots$
give a $D_{s}$-module isomorphism

$\psi_{0}:N\to\phi^{r}(N)$

for some $r$ dividing $s$ with $0<r<s$. Let
$\psi_{i}=\phi^{ir}\circ\psi_{0}\circ\phi^{-ir}:\phi^{ir}(N)\rightarrow\phi^{(i+1)r}(N)$.
Then $c\coloneqq\psi_{s/r-1}\circ\ldots\circ\psi_{1}\circ\psi_{0}$ is in
$\operatorname{End}_{D_{s}}(N)=C$. After rescaling each $\psi_{i}$ by the
factor ${c}^{-r/s}$, we may assume $c=1$. We further define

$\Psi_{i}=\psi_{i-1}\circ\cdots\circ\psi_{0}\colon N\to\phi^{ir}(N)$

and

$\Psi\colon N\to M\colon\quad n\mapsto\sum_{i=0}^{s/r-1}\Psi_{i}(n).$

Since $\Psi_{1}=\psi_{0}\colon N\to\phi^{r}(N)$ is an isomorphism, if $n\in N$
then $\phi^{r}(n)=\Psi_{1}(n^{\prime})$ for some $n^{\prime}\in N$. From
$\Psi_{s/r}=c=1=\Psi_{0}$ we get

$\phi^{r}\left(\Psi(n)\right)=\sum_{i=0}^{s/r-1}\phi^{r}\Psi_{i}(n)=\sum_{i=0}^{s/r-1}\Psi_{i+1}(n^{\prime})=\Psi(n^{\prime})$

which shows that $\Psi(N)\subsetneq M\mathord{\downarrow}^{1}_{r}$ is a
$D_{r}$-module. Thus $M\mathord{\downarrow}^{1}_{r}$ is reducible,
contradicting the minimality of $s$. ∎

The following corollary is used in [2] to detect restriction-reducibility.

######  Corollary 6.6.

If $M$ is restriction-reducible, then $M\mathord{\downarrow}^{1}_{p}$ is
reducible for some prime factor $p$ of $\dim_{F}M$.

###### Proof.

By Theorem 6.5, $M\cong N\mathord{\uparrow}_{t}^{1}$ for some $D_{t}$-module
$N$ with $t>1$. Let $p$ be a prime factor of $t$ and let
$P=N\mathord{\uparrow}_{t}^{p}$. Then $M\cong P\mathord{\uparrow}_{p}^{1}$ so
$M\mathord{\downarrow}^{1}_{p}\cong\bigoplus_{i=0}^{p-1}\phi^{i}(P)$ is
reducible, and $p$ divides
$\dim_{F}M=\dim_{F}M\mathord{\downarrow}^{1}_{p}=p\cdot\dim_{F}P$. ∎

The rest of this section is not needed for Section 7. Our goal here is to give
a more precise decomposition in Proposition 6.2 by using Theorem 6.5 and the
following analogue of [10, Theorem 22.2].

######  Theorem 6.7 (Mackey’s formula).

Let $s,t\geq 1$, $d=\gcd(s,t)$, and $l=\operatorname{lcm}(s,t)$. Given a
$D_{t}$-module $N$, we have a $D_{s}$-module isomorphism

$N\mathord{\uparrow}^{1}_{t}\mathord{\downarrow}^{1}_{s}\cong\bigoplus_{i=0}^{d-1}\left(\phi^{i}\otimes
N\right)\mathord{\downarrow}^{t}_{l}\mathord{\uparrow}^{s}_{l}.$ (4)

###### Proof.

We have isomorphisms of $D_{s}$-modules (of infinite $F$-dimension)

$D\cong\bigoplus_{i=0}^{d-1}D_{d}\otimes\phi^{i}\cong\bigoplus_{i=0}^{d-1}D_{s}\otimes_{D_{l}}(\phi^{i}\otimes
D_{t})$

where the last isomorphism sends $\phi^{js+kt}\otimes\phi^{i}$ to
$\phi^{js}\otimes\phi^{i}\otimes\phi^{kt}$. Therefore

$\begin{split}N\mathord{\uparrow}^{1}_{t}\mathord{\downarrow}^{1}_{s}&\cong\left(D\otimes_{D_{t}}N\right)\mathord{\downarrow}^{1}_{s}\\\
&\cong\bigoplus_{i=0}^{d-1}D_{s}\otimes_{D_{l}}\left(\phi^{i}\otimes
D_{t}\right)\otimes_{D_{t}}N\\\
&\cong\bigoplus_{i=0}^{d-1}D_{s}\otimes_{D_{l}}\left(\phi^{i}\otimes
N\right)\mathord{\downarrow}^{t}_{l}\\\
&\cong\bigoplus_{i=0}^{d-1}\left(\phi^{i}\otimes
N\right)\mathord{\downarrow}^{t}_{l}\mathord{\uparrow}^{s}_{l}.\end{split}$

∎

######  Corollary 6.8.

If $N$ is absolutely irreducible, then the direct summands
$\left(\phi^{i}\otimes
N\right)\mathord{\downarrow}^{t}_{l}\mathord{\uparrow}^{s}_{l}$ in (4) are
irreducible and induced from the absolutely irreducible modules
$\phi^{i}\otimes N\mathord{\downarrow}^{t}_{l}$.

###### Proof.

Observe that $(\phi^{i}\otimes
N)\mathord{\downarrow}^{t}_{l}\,\mathord{\uparrow}^{s}_{l}$ is a direct sum
(as $D_{l}$-modules) of $(\phi^{j}\otimes N)\mathord{\downarrow}^{t}_{l}$ for
$j$ in

$I\coloneqq\left\\{i+ks\pmod{t}\mid 0\leq k<\frac{l}{s}\right\\}.$

Since $N$ is absolutely irreducible, so is $(\phi^{j}\otimes
N)\mathord{\downarrow}^{t}_{l}$. The smallest $D_{s}$-module containing
$(\phi^{j}\otimes N)\mathord{\downarrow}^{t}_{l}$ is $(\phi^{i}\otimes
N)\mathord{\downarrow}^{t}_{l}\,\mathord{\uparrow}^{s}_{l}$ as $s\mathbb{Z}$
acts transitively on $I$ by addition. Therefore $(\phi^{i}\otimes
N)\mathord{\uparrow}^{s}_{l}$ is irreducible. ∎

##  7\. Main theorem

We now prove our main result by reducing to the essential case where the total
Picard-Vessiot ring is a field in Lemma 7.2.

######  Theorem 7.1.

Let $(F,\phi)=(C(x),x\mapsto x+1)$. Let $M$ be a $D$-module of dimension $3$.
If a nonzero $\mathcal{S}$-point of $M$ is $2$-expressible, then

  1. (1)

$M$ is reducible over $F$, or

  2. (2)

$M\cong N\mathord{\uparrow}^{1}_{3}$ for some $D_{3}$-module $N$, or

  3. (3)

$M\cong S^{2}(N)\otimes P$ for some $D$-submodules $N$ and $P$ of dimensions
$2$ and $1$, respectively.

We can now deduce the restatement of Theorem 7.1 in Section 1: let $L$ be a
$2$-solvable operator of order 3, so that Theorem 7.1 applies to $M_{L}$. If
$M_{L}$ is reducible, so is $L$. If $M\cong N\mathord{\uparrow}^{1}_{3}$, then
$L$ is gauge equivalent to some $\phi^{3}+f$ per the discussion in Section 5.
Finally suppose that $M_{L}\cong S^{2}(N)\otimes P$ and $L_{2}$ and $L_{1}$
are minimal operators of cyclic vectors for $N$ and $P$, respectively. If
$L_{2}^{\mathbin{\circledS}2}$ has order $2$, then $L$ is reducible. Otherwise
$L_{2}^{\mathbin{\circledS}2}$ has order $3$ and $L$ is gauge equivalent to
$L_{2}^{\mathbin{\circledS}2}\mathbin{\circledS}L_{1}$.

###### Proof of Theorem 7.1.

If $M$ is reducible, (1) holds. If $M$ is restriction-reducible, then $M\cong
N\mathord{\uparrow}^{1}_{t}$ for some $t>1$. A dimension count gives $t=3$, so
(2) holds.

We may now assume $M$ is absolutely irreducible. Since $L$ is $2$-solvable,
$\operatorname{Gal}(L)$ is Eulerian by Theorem 3.2. By Proposition 5.2 (2),
there exists $t\geq 1$ for which the total Picard-Vessiot ring of
$M\mathord{\downarrow}^{1}_{t}$ is a difference field $(K,\phi^{t})$ and
$\operatorname{Gal}(M\mathord{\downarrow}^{1}_{t})=\operatorname{Gal}(M)^{\circ}$.
Thus $\operatorname{Gal}(M\mathord{\downarrow}_{t}^{1})$ is Eulerian and
connected.

Let $S^{2}(M)=Q_{1}\oplus Q_{2}\oplus\cdots\oplus Q_{r}$ be an irreducible
decomposition of $S^{2}(M)$. Then

$S^{2}(M\mathord{\downarrow}_{t}^{1})=Q_{1}\mathord{\downarrow}^{1}_{t}\oplus
Q_{2}\mathord{\downarrow}^{1}_{t}\oplus\cdots\oplus
Q_{r}\mathord{\downarrow}^{1}_{t}.$

By Proposition 6.2, $Q_{i}\mathord{\downarrow}^{1}_{t}$ is a direct sum of
irreducible $D_{t}$-modules of the same dimension. Lemma 7.2 below now
requires $r=2$, and (up to labeling)
$\dim_{F}Q_{1}\mathord{\downarrow}^{1}_{t}=1$, and
$\dim_{F}Q_{2}\mathord{\downarrow}^{1}_{t}=5$. In particular,
$\dim_{F}Q_{1}=1$. Finally, Proposition 4.1 gives $M\cong S^{2}(N)\otimes P$
for some $D$-submodules $N$ and $P$ of dimensions $2$ and $1$, respectively. ∎

To finish the proof of Theorem 7.1, we need

######  Lemma 7.2.

Let $(F,\phi)=(C(x),x\mapsto x+1)$. Let $M$ be an absolutely irreducible
$D$-module of dimension $3$ whose total Picard-Vessiot ring is a field and
whose Galois group is Eulerian and connected. Then $S^{2}(M)$ is the direct
sum of two irreducible $D$-modules of dimensions $1$ and $5$, respectively.

###### Proof.

$G\coloneqq\operatorname{Gal}(M/F)$ embeds into $\operatorname{GL}_{3}(C)$
since $\dim_{F}M=3$. Because $G$ is connected, the group homomorphism

$G\hookrightarrow\operatorname{GL}_{3}(C)\xrightarrow{\det}\mathbb{G}_{m}$

has a kernel $H\leq\operatorname{SL}_{3}(C)$ and image $1$ or
$\mathbb{G}_{m}$.

Let $K$ be the total Picard-Vessiot ring of $M$ over $F$,
$K^{\prime}=K^{H^{\circ}}$, and $K^{\prime\prime}=K^{H}$. Then $K/F$
decomposes as

${K}$${K^{\prime}}$${K^{\prime\prime}}$${F}$$\scriptstyle{H^{\circ}}$$\scriptstyle{H}$$\scriptstyle{G}$finite$\scriptstyle{1\text{
or }\mathbb{G}_{m}}$

First we show $K^{\prime}/F$ is a total Picard-Vessiot extension with Galois
group $G/H^{\circ}$ and $G/H^{\circ}$ is either $1$ or $\mathbb{G}_{m}$. Since
$H^{\circ}$ is characteristic in $H$ and $H\trianglelefteq G$, we have
$H^{\circ}\trianglelefteq G$ and $K^{\prime}/F$ is a total Picard-Vessiot
extension with Galois group $G/H^{\circ}$. As a quotient of $G$, $G/H^{\circ}$
is connected. If $G/H=1$, then $G/H^{\circ}$ is finite and connected, hence
$1$. If instead $G/H=\mathbb{G}_{m}$, then $G/H^{\circ}$ is connected of
dimension $1$ with quotient $G/H=\mathbb{G}_{m}$. The only connected algebraic
groups of dimension $1$ are $\mathbb{G}_{a}$ and $\mathbb{G}_{m}$ [20, Theorem
3.4.9], so $G/H^{\circ}=\mathbb{G}_{m}$.

Note that $H^{\circ}\leq\operatorname{SL}_{3}(C)$ is Eulerian. By [18,
Proposition 4.1], the possibilities for $H^{\circ}$ are:

  1. (a)

$H^{\circ}$ is finite; or

  2. (b)

$V(M)$ has an $H^{\circ}$-submodule of dimension $2$; or

  3. (c)

$V(M)$ has an $H^{\circ}$-submodule of dimension $1$; or

  4. (d)

$H^{\circ}\cong\operatorname{PSL}_{2}(C)$, and $V(M)\cong R_{2}$ as
$\operatorname{PSL}_{2}(C)$-modules. Here $R_{2}$ is the degree $2$
homogeneous part of $C[x,y]$, equipped with the standard action of
$\operatorname{SL}_{2}(C)$ on degree $1$ and extended to all degrees.

Since $M$ is absolutely irreducible, the contrapositive of Proposition 5.2(1)
shows that $G$ is not solvable. To conclude the proof, we will show that Cases
(a)–(c) force $H^{\circ}$ (and hence $G$) to be solvable and so cannot occur.
We will show that Case (d) gives our desired result.

Case (a). We must have $H^{\circ}=1$ (solvable). This rules out Case (a).

Case (b). Since $M$ is irreducible, an $H^{\circ}$-submodule $U\subset V(M)$
of dimension $2$ is not $G$-stable by Proposition 2.1. There exists $g\in G$
such that $U\cap g(U)\subset V(M)$ is an $H^{\circ}$-submodule since
$H^{\circ}\trianglelefteq G$. By dimension reasons, $\dim_{F}U\cap g(U)=1$,
which reduces to Case (c).

Case (c). Let $U\subset V(M)$ be a $H^{\circ}$-submodule of dimension $1$. The
restriction of the irreducible $G$-representation $V(M)$ to
$H^{\circ}\trianglelefteq G$ is a semisimple representation $V(M)=\bigoplus
g(U)$ over some $g\in G$. Therefore
$\operatorname{Gal}(K/K^{\prime})=H^{\circ}\leq\mathbb{G}_{m}^{3}$ is
solvable, ruling out Case (c).

Case (d). From $V(M)\cong R_{2}$, we get

$V(S^{2}(M))\cong S^{2}(V(M))\cong S^{2}(R_{2}).$

The $\operatorname{PSL}_{2}(C)$-module $S^{2}(R_{2})$ is known to be a direct
sum of irreducible modules of dimensions $1$ and $5$. Thus $V(S^{2}(M))$
decomposes in the same way as $H^{\circ}$-modules and as $G$-modules (since
the direct summands have different dimensions). By Proposition 2.1, $S^{2}(M)$
has irreducible submodules of dimensions $1$ and $5$. Since $S^{2}(M)$ is
semisimple, this is a direct sum. ∎

##  8\. Sectioning and interlacing

In this section, we show how restricted and induced modules give coordinate-
free formulation of sectioning and interlacing of sequences. This is first
done on sequences and then on solutions of the modules. We will assume
$F\subseteq\mathcal{S}$ so that every difference module has a cyclic vector.

###  8.1. Sequences

Recall from [3, Definition 3.2] that the $i$th $t$-_section_ of a sequence $s$
is the sequence $s^{i}_{t}$ defined by $s^{i}_{t}(j)=s(j)$ if $j\equiv
i\pmod{t}$ and $s^{i}_{t}(j)=0$ otherwise. The set $\mathcal{S}^{i}_{t}$ of
$i$th $t$-sections of $s\in\mathcal{S}$ is then a $D$-algebra where $F$ acts
by multiplication within $\mathcal{S}$. The maps

$\displaystyle\varphi\colon$
$\displaystyle\mathcal{S}\mathord{\downarrow}^{1}_{t}\longrightarrow\mathcal{S}^{0}_{t}\mathord{\uparrow}^{1}_{t}\mathord{\downarrow}^{1}_{t}\colon$
$\displaystyle s$
$\displaystyle\longmapsto{\textstyle\sum}_{i}\,\phi^{i}\otimes s^{i}_{t}$
$\displaystyle\psi\colon$
$\displaystyle\mathcal{S}^{0}_{t}\mathord{\uparrow}^{1}_{t}\longrightarrow\mathcal{S}\colon$
$\displaystyle{\textstyle\sum}_{i}\,\phi^{i}\otimes s^{i}$
$\displaystyle\longmapsto{\textstyle\sum}_{i}\,\phi^{i}(s^{i})$

are respectively $D_{t}$\- and $D$-algebra isomorphisms that split and combine
a sequence with its $t$-sections. We can also identify
$\mathcal{S}^{0}_{t}\mathord{\uparrow}^{1}_{t}\mathord{\downarrow}^{1}_{t}$
with $\prod_{i=0}^{t}\mathcal{S}^{i}_{t}$ by taking $\sum\phi^{i}\otimes
s^{i}$ to $(s^{0}_{t},s^{1}_{t},\dots,s^{t-1}_{t})$. This explains restriction
and induction of sequences.

###  8.2. Operators

To explain solutions of restricted operators, we need

######  Proposition 8.1.

Let $L\in F[\phi]$. Then
$L\mathord{\downarrow}^{1}_{t}=L\mathbin{\circledS}(\phi^{t}-1)$.

###### Proof.

Let $\zeta_{t}$ be a primitive $t$-th root of unity and let
$L^{\prime}=L\mathbin{\circledS}(\phi^{t}-1)$, the minimal operator in $D$ of
$1\otimes 1\in M_{L}\otimes M_{\phi^{t}-1}$. As $L^{\prime}$ is invariant
under $\sigma\colon\phi\mapsto\zeta_{t}\phi$ by the computation

$\sigma(L^{\prime})=L^{\prime}\mathbin{\circledS}(\zeta_{t}\phi-1)=L\mathbin{\circledS}(\phi^{t}-1)\mathbin{\circledS}(\zeta_{t}\phi-1)=L\mathbin{\circledS}(\phi^{t}-1)=L^{\prime},$

we have $L^{\prime}\in D_{t}$, so $L^{\prime}$ is the minimal operator of
$1\otimes 1\in(M_{L}\otimes M_{\phi^{t}-1})\mathord{\downarrow}^{1}_{t}$. But
$L\mathord{\downarrow}^{1}_{t}$ is already such a minimal operator since the
map

$M_{L}\to M_{L}\otimes M_{\phi^{t}-1}\colon m\mapsto m\otimes 1$

is a $D_{t}$-module embedding. Therefore
$L\mathord{\downarrow}^{1}_{t}=L^{\prime}$. ∎

Let $e^{i}_{t}$ be the $i$th $t$-section of $e=(1,1,\dots)$. For $R\in D_{t}$
and $L\in D$, the relations
$V_{\mathcal{S}}(R)e^{i}_{t}=V_{\mathcal{S}^{i}_{t}}(R)$ and
$L\mathord{\downarrow}^{1}_{t}=L\mathbin{\circledS}(\phi^{t}-1)$ give

$V_{\mathcal{S}}(R)=\bigoplus_{i=0}^{t-1}V_{\mathcal{S}^{i}_{t}}(R),\quad
V_{\mathcal{S}}(L\mathord{\downarrow}^{1}_{t})=\bigoplus_{i=0}^{t-1}V_{\mathcal{S}}(L)e^{i}_{t},\quad
V_{\mathcal{S}^{i}_{t}}(L\mathord{\downarrow}^{1}_{t})=V_{\mathcal{S}}(L)e^{i}_{t}.$

###  8.3. Modules

The above result in terms of modules read as follows: For a $D$-module $M$ and
a $D_{t}$-module $N$, the maps

$\displaystyle\bigoplus_{i=0}^{t-1}V_{\mathcal{S}^{i}_{t}}(N)$
$\displaystyle\longrightarrow
V_{\mathcal{S}}(N\mathord{\uparrow}^{1}_{t})\colon$
$\displaystyle(s^{i})_{i=0}^{t-1}$
$\displaystyle\longmapsto{\textstyle\sum_{i}}s^{i}$ $\displaystyle
V_{\mathcal{S}}(M)$ $\displaystyle\longrightarrow
V_{\mathcal{S}^{i}_{t}}(M\mathord{\downarrow}^{1}_{t})\colon$ $\displaystyle
s$ $\displaystyle\longmapsto s^{i}_{t}$

are respectively bijective and surjective.

###  8.4. Fractional induction and interlacing

We define $C[\phi^{\pm 1/t}]$ as the formal polynomial ring $C[\phi^{1/t}]$
localized by $\phi^{1/t}$. Then

$F\mathord{\uparrow}^{1/t}_{1}\coloneqq C[\phi^{\pm 1/t}]\otimes_{C[\phi^{\pm
1}]}F$

can be identified with $F^{t}$ in the $F$-basis $\\{\phi^{-i/t}\otimes
1\\}_{i=0}^{t-1}$. In these coordinates, $\phi^{1/t}$ acts on $F^{t}$ as

$\phi^{1/t}\cdot(f_{0},f_{1},f_{2}\dots,f_{t-1})=(f_{1},f_{2}\dots,f_{t-1},\phi(f_{0})).$

Componentwise addition and multiplication in $F^{t}$ makes
$F\mathord{\uparrow}^{1/t}_{1}$ a difference ring. We also define
$D_{1/t}\coloneqq F\mathord{\uparrow}^{1/t}[\phi^{\pm 1/t}]$ and the
$D_{1/t}$-module $M\mathord{\uparrow}^{1/t}\coloneqq D_{1/t}\otimes_{D}M$.

######  Example 8.2 (Interlacing of sequences).

Writing $\mathcal{S}\mathord{\uparrow}^{1/t}_{1}$ as
$(\mathcal{S}^{t},\phi^{1/t})$, the map $\mathcal{S}^{t}\to\mathcal{S}$ taking
$(s^{0},s^{1},\dots,s^{t-1})$ to their interlacing

$(s^{0}(0),s^{1}(0),\dots,s^{t-1}(0),s^{0}(1),s^{1}(1),\dots,s^{t-1}(1),\dots).$

is a difference ring isomorphism. Thus
$\mathcal{S}\mathord{\uparrow}^{1/t}_{1}$ corresponds to interlacing.

##  Appendix A A Gröbner basis computation

We now verify a computation in the proof of Proposition 4.1. The list I below
contains the relations (2), as well as $1-\det(g_{ij})\cdot z$ which forces
$\det(g_{ij})$ to be nonzero (Rabinowitsch trick). Next, I_groebner is the
Gröbner basis of I; G and A are the left and right sides of (3); J lists the
relations (3) after scaling by $4g_{11}g_{21}^{2}$ to clear denominators;
IJ_groebner is the Gröbner basis of the entries of J concatenated with
I_groebner. Since line 15 returns True, the relations in J are implied by
those in I.

⬇

import sympy

from sympy import symbols, groebner

from sympy.matrices import Matrix

\para11, a12, a21, a22, g11, g12, g13, g21, g22, g23, g31, g32, g33, z =
symbols(’a11 a12 a21 a22 g11 g12 g13 g21 g22 g23 g31 g32 g33 z’)

\parG = Matrix([[g11, g12, g13], [g21, g22, g23], [g31, g32, g33]])

A = g11 * Matrix([[a11**2, 2*a11*a12, a12**2], [a11*a21, a11*a22+a12*a21,
a12*a22], [a21**2, 2*a21*a22, a22**2]])

\parI = [1-G.det()*z, g21**2-g11*g31, g23**2-g13*g33,
2*g21*g22-g11*g32-g31*g12, 2*g22*g23-g12*g33-g32*g13, g22**2-g12*g32 +
2*g21*g23-g11*g33-g31*g13]

I_groebner = groebner(I, g11, g12, g13, g21, g22, g23, g31, g32, g33, z)

\parJ = (G - A.subs([(a11, 1), (a12, g12/(2*g11)), (a21, g21/g11), (a22,
g32/(2*g21))])) * 4 * g11 * g21**2

IJ_groebner = groebner(J[:][:] + I_groebner[:], g11, g12, g13, g21, g22, g23,
g31, g32, g33, z)

IJ_groebner == I_groebner # returns True

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