Local and global canonical height functions for affine space regular automorphisms

TL;DR

This paper constructs local and global canonical height functions for affine space automorphisms over number fields, providing tools to analyze the arithmetic dynamics of these automorphisms and their points.

Contribution

It introduces v-adic Green functions for automorphisms and their inverses, and defines a notion of good reduction to establish a global canonical height with finiteness properties.

Findings

Constructed v-adic Green functions for automorphisms and their inverses.

Defined good reduction at places to build a global canonical height.

Provided height growth estimates under automorphisms and their inverses.

Abstract

Let f: A^N \to A^N be a regular polynomial automorphism defined over a number field K. For each place v of K, we construct the v-adic Green functions G_{f,v} and G_{f^{-1},v} (i.e., the v-adic canonical height functions) for f and f^{-1}. Next we introduce for f the notion of good reduction at v, and using this notion, we show that the sum of v-adic Green functions over all v gives rise to a canonical height function for f that satisfies the Northcott-type finiteness property. Using previous results, we recover results on arithmetic properties of f-periodic points and non f-periodic points. We also obtain an estimate of growth of heights under f and f^{-1}, which is independently obtained by Lee by a different method.

Equations

$$\underset{n\to +\infty }{\lim }\frac{1}{d^n}\log \max \{\Vert f^n(x)\Vert ,1\}\text{and}\underset{n\to +\infty }{\lim }\frac{1}{d_{-}^n}\log \max \{\Vert f^{-n}(x)\Vert ,1\}$$

$$G_f(\cdot )\le \log \max \{\Vert \cdot \Vert ,1\}+c_f\text{and}{G}_{f^{-1}}(\cdot )\le \log \max \{\Vert \cdot \Vert ,1\}+c_{f^{-1}}\text{on AN(Ω)}.$$

$$G_f(\cdot )$$

$$G_{f^{-}}(\cdot )$$

$${\hat{h}}_f^{+}(x):=\underset{n\to +\infty }{\lim }\frac{1}{d^n}h(f^n(x))\text{and}{\hat{h}}_f^{-}(x):=\underset{n\to +\infty }{\lim }\frac{1}{d_{-}^n}h(f^{-n}(x))$$

$${\hat{h}}_f^{+}(x)=\sum _{v\in M_K}{n}_v{G}_{f,v}(x)\text{and}{\hat{h}}_f^{-}(x)=\sum _{v\in M_K}{n}_v{G}_{f^{-1},v}(x).$$

$$\frac{1}{d}{\hat{h}}_f\circ f+\frac{1}{d_{-}}{\hat{h}}_f\circ f^{-1}=\left(1+\frac{1}{d{d}_{-}}\right){\hat{h}}_f.$$

$${\hat{h}}_f(x)=0⟺{\hat{h}}_f^{+}(x)=0⟺{\hat{h}}_f^{-}(x)=0⟺\text{x is f-periodic.}$$

$$\frac{1}{d}h(f(x))+\frac{1}{d_{-}}h(f^{-1}(x))\ge \left(1+\frac{1}{d{d}_{-}}\right)h(x)-c$$

$$\underset{\begin{array}{c}x\in {\mathbb{A}}^N(\overline{K})\\ h(x)\to \infty \end{array}}{\mathrm{liminf}}\frac{\frac{1}{d}h(f(x))+\frac{1}{d_{-}}h(f^{-1}(x))}{h(x)}=1+\frac{1}{d{d}_{-}}.$$

$$\#\{y\in O_f(x)\mid h(y)\le T\}=\left(\frac{1}{\log d}+\frac{1}{\log d_{-}}\right)\log T-\hat{h}(O_f(x))+O(1)$$

$$a_n:=\frac{1}{d^n}{\log }^{+}\Vert f^n(x)\Vert ,b_n:=\frac{1}{d^n}\log \Vert F^n(x,1)\Vert ,c_n:=\frac{1}{d^n}\log \Vert (rF{)}^n(x,1)\Vert ,$$

$$a_n=b_n=c_n-\frac{1-d^{-n}}{d-1}\log |r|.$$

$$a_n$$

$$=\frac{1}{d^n}\log \max \{|F_{n1}(x,1)|,\dots ,|F_{nN}(x,1)|,1\}$$

$$=\frac{1}{d^n}\log \max \{|F_1^n(x,1)|,\dots ,|F_N^n(x,1)|,1\}=b_n.$$

$$\frac{1}{d^n}\log \Vert (rF{)}^n(x,1)\Vert \le \frac{1}{d^n}\log \Vert (rF{)}^{n-1}(x,1){\Vert }^d\le \frac{1}{d^{n-1}}\log \Vert (rF{)}^{n-1}(x,1)\Vert .$$

$$G_f(x):=\underset{n\to +\infty }{\lim }\frac{1}{d^n}{\log }^{+}\Vert f^n(x)\Vert \text{for x∈AN(Ω)},$$

$$G_f(\cdot )\le {\log }^{+}\Vert \cdot \Vert +c_f\text{on AN(Ω)}.$$

$$G_f(x)\le \underset{n\to +\infty }{\lim }{c}_n-\frac{1}{d-1}\log |r|\le c_0-\frac{1}{d-1}\log |r|={\log }^{+}\Vert x\Vert -\frac{1}{d-1}\log |r|.$$

$$\frac{1}{d^{n_{0}m}}{\log }^{+}\Vert f^{n_{0}m}(x)\Vert ={\left(\frac{d_{n_0}}{d^{n_0}}\right)}^m\frac{1}{d_{n_0}^m}{\log }^{+}\Vert g^m(x)\Vert ⟶0\text{(m→+∞)}.$$

$$I_{+}=\{(x:0)\in {\mathbb{P}}^N(\Omega )\mid F_1(x,0)=\cdots =F_N(x,0)=0\},$$

$$I_{-}=\{(x:0)\in {\mathbb{P}}^N(\Omega )\mid G_1(x,0)=\cdots =G_N(x,0)=0\}.$$

$$\sum _{j=1}^N{P}_{ij}(X)F_j(X,0)+\sum _{j=1}^N{Q}_{ij}(X)G_j(X,0)=X_i^m$$

$$\sum _{j=1}^N{P}_{ij}(X)F_j(X,T)+\sum _{j=1}^N{Q}_{ij}(X)G_j(X,T)+T{R}_i(X,T)=X_i^m.$$

$$C=\max \{C^{\prime },1\}.$$

$$\epsilon \le \frac{1}{C},\delta \le \frac{1}{C},(\epsilon C{)}^d\le \delta ,\text{and}(\epsilon C{)}^{d_{-}}\le \delta .$$

$$\epsilon =\frac{1}{C^{\min \{d,d_{-}\}}}\text{and}\delta =\frac{1}{C^{\min \{d,d_{-}\}(\min \{d,d_{-}\}-1)}}.$$

$$N_{\delta ,\epsilon }^{+}$$

$$V_{\delta ,\epsilon }^{+}$$

$${\overline{N}}_{\delta ,\epsilon }^{+}=\left\{(x:t)\in {\mathbb{P}}^N(\Omega )||t|<\epsilon \Vert x\Vert \text{and}\Vert (F(x,t),t^d)\Vert <\delta \Vert (x,t){\Vert }^d\right\}.$$

$$I_{+}\subseteq {\overline{N}}_{\delta ,\epsilon }^{+}.$$

$$f^{-1}(N_{\delta ,\epsilon }^{+})\subseteq N_{\delta ,\epsilon }^{+}.$$

Full text

2000 Mathematics Subject Classification: Primary: 11G50, Secondary: 32H50

Key words and phrases. canonical height, local canonical height, regular polynomial automorphism

Supported in part by MEXT grant-in-aid for young scientists (B) 21740018

Introduction

Height functions are one of the basic tools in Diophantine geometry. On Abelian varieties defined over a number field, there exist Néron–Tate’s canonical height functions that behave well relative to the $[n]$-th power map. Tate’s elegant construction is via a global method using a relation of an ample divisor relative to the $[n]$-th power map. Néron’s construction is via a local method, and gives deeper properties of the canonical height functions. Both constructions are useful in studying arithmetic properties of Abelian varieties.

In [7], we showed the existence of canonical height functions for affine plane polynomial automorphisms of dynamical degree $\geq 2$. Our construction was via a global method using the effectiveness of a certain divisor on a certain rational surface. In this paper, we use a local method to construct a canonical height function for affine space regular automorphism $f:{\mathbb{A}}^{N}\to{\mathbb{A}}^{N}$, which coincides with the one in [7] when $N=2$. We note that arithmetic properties of polynomial automorphisms over number fields have been studied, for example, by Silverman [17], Denis [5], Marcello [12, 13] and the author [7].

We recall the definition of regular polynomial automorphisms. Let $f:{\mathbb{A}}^{N}\to{\mathbb{A}}^{N}$ be a polynomial automorphism of degree $d\geq 2$ defined over a field, and $\overline{f}:{\mathbb{P}}^{N}\dasharrow{\mathbb{P}}^{N}$ denote its birational extension to ${\mathbb{P}}^{N}$. We write $f^{-1}$ for the inverse of $f$, $d_{-}$ for the degree of $f^{-1}$, and $\overline{f^{-1}}$ for its birational extension to ${\mathbb{P}}^{N}$. Then $f$ is said to be regular if the intersection of the set of indeterminacy of $\overline{f}$ and that of $\overline{f^{-1}}$ is empty over an algebraic closure of the field (cf. Definition 2.1 and Remark 2.2). Over ${\mathbb{C}}$, dynamical properties of affine space regular polynomial automorphisms $f$ are deeply studied, in which the Green function for $f$ plays a pivotal role (see [16, §2]).

In §1 and §2, we construct a Green function (a local canonical height function) for $f$ over an algebraically closed field $\Omega$ with non-trivial non-archimedean absolute value $|\cdot|$. For $x=(x_{1},\ldots,x_{N})\in\Omega^{N}$, we set $\|x\|=\max_{1\leq i\leq N}\{|x_{i}|\}$. Our results are put together as follows.

Theorem A** (cf. Proposition 1.1, Lemma 1.3, Theorem 2.3).**

Let $f:{\mathbb{A}}^{N}\to{\mathbb{A}}^{N}$ be a regular polynomial automorphism of degree $d\geq 2$ defined over $\Omega$. With the notation as above, we have the following.

1. (1)

For all $x\in{\mathbb{A}}^{N}(\Omega)$, the limits

[TABLE]

exist. We respectively write $G_{f}(x)$ and $G_{f^{-1}}(x)$ for the limits, which we call Green functions for $f$ and $f^{-1}$. 2. (2)

There are constants $c_{f},c_{f^{-1}}\in{\mathbb{R}}$ such that

[TABLE] 3. (3)

There are subsets $V^{+},V^{-}$ of ${\mathbb{A}}^{N}(\Omega)$ with $V^{+}\cup V^{-}={\mathbb{A}}^{N}(\Omega)$ and constants $c^{+},c^{-}\in{\mathbb{R}}$ such that

[TABLE]

Over ${\mathbb{C}}$, Green functions are constructed using compactness arguments (cf. [16, §2]). Here we use more algebraic arguments based on Hilbert’s Nullstellensatz. Our construction of $V^{\pm},c^{\pm}$ is rather delicate with a choice of two parameters $\varepsilon$ and $\delta$, which behaves well when we work over number fields in §6 and §7. We note that, over ${\mathbb{C}}$, our construction gives a different proof of the existence of Green functions with more explicit estimates (see §5). In §3, we continue to study some basic properties of regular polynomial automorphisms $f$ over $\Omega$, characterizing the set of escaping points by $G_{f}$ and showing a filtration property for $f$.

Now we turn our attention to number fields. Let $f:{\mathbb{A}}^{N}\to{\mathbb{A}}^{N}$ be a polynomial automorphism defined over a number field $K$. For each place $v$ of $K$, let $K_{v}$ denote the completion of $K$ with respect to $v$, and $\overline{K_{v}}$ an algebraic closure of $K_{v}$. Then $f$ induces a regular polynomial automorphism over $\overline{K_{v}}$, so we have Green functions $G_{f,v},G_{f^{-1},v}$ and estimates with $c_{f,v},c_{f^{-1},v},c^{\pm}_{v}$ as in Theorem A. (Here we use the suffix $v$ to indicate that we work over $\overline{K_{v}}$. See §5 when $v$ is Archimedean.)

We want to define the canonical height functions $\widehat{h}_{f}^{+},\widehat{h}_{f}^{-}$ for $f$ as the sum of $G_{f,v},G_{f^{-1},v}$ over all the places $v$ of $K$. To this end, we introduce the notion of good reduction at an non-Archimedean place $v$ of $K$. Let $\overline{R_{v}}$ denote the ring of integers of $\overline{K_{v}}$, and $\widetilde{k_{v}}$ the residue field. Recall that the notion of good reduction for an endomorphism $\varphi$ of ${\mathbb{P}}^{1}$ over $\overline{K_{v}}$ is introduced in Morton–Silverman [14], which means that $\varphi$ extends to a morphism over $\overline{R_{v}}$ and the induced morphism $\widetilde{\varphi}$ over $\widetilde{k_{v}}$ has the same degree as $\varphi$. Here we say that a regular polynomial automorphism $f:{\mathbb{A}}^{N}\to{\mathbb{A}}^{N}$ has good reduction at $v$ if $f$ extends to an automorphism over $\overline{R_{v}}$ and the the induced morphism $\widetilde{f}$ over $\widetilde{k_{v}}$ is again a regular polynomial automorphism such that the degrees of $\widetilde{f}$ and $\widetilde{f}^{-1}$ are the same as the degrees of $f$ and $f^{-1}$ respectively (see Definition 4.1 for the precise definition).

Using the notion of good reduction, we show the existence of canonical height functions. Let $h:{\mathbb{A}}^{N}(\overline{K})\to{\mathbb{R}}$ denote the usual logarithmic Weil height function.

Theorem B** (cf. Proposition 6.2, Theorem 6.3).**

Let $f:{\mathbb{A}}^{N}\to{\mathbb{A}}^{N}$ be a regular polynomial automorphism of degree $d\geq 2$ over a number field $K$. Let $d_{-}\geq 2$ denote the degree of $f^{-1}$.

1. (1)

$f$* has good reduction at $v$ except for finitely many places. Further, if this is the case, we can take the constants $c_{f,v}=c_{f^{-1},v}=c^{\pm}_{v}=0$ in Theorem A.* 2. (2)

For all $x\in{\mathbb{A}}^{N}(\overline{K})$, the limits

[TABLE]

exist. Further, we have the decomposition into the sum of local Green functions:**

[TABLE] 3. (3)

We define $\widehat{h}_{f}:{\mathbb{A}}^{N}(\overline{K})\to{\mathbb{R}}$ by $\widehat{h}_{f}:=\widehat{h}^{+}_{f}+\widehat{h}^{-}_{f}$. Then $\widehat{h}_{f}$ satisfies $\widehat{h}_{f}\gg\ll h$ and

[TABLE]

Further, for $x\in{\mathbb{A}}^{N}(\overline{K})$, we have

[TABLE]

We note that in [7], we have defined $\widehat{h}^{+}_{f}(x)$ as $\limsup_{n\to\infty}\frac{1}{d^{n}}h(f^{n}(x))$, and similarly for $\widehat{h}^{-}_{f}$. Theorem B shows that $\{\frac{1}{d^{n}}h(f^{n}(x))\}_{n=0}^{+\infty}$ and $\{\frac{1}{d_{-}^{n}}h(f^{-n}(x))\}_{n=0}^{+\infty}$ are in fact convergent sequences, i.e., $\limsup$ can be replaced by $\lim$ in (0.1).

Using estimates on local Green functions over all places, we obtain the following estimate on global canonical height functions for all $N\geq 2$ (see [7, §4], [18, Conjecture 3], [19, Conjecture 7.18]). This result is obtained by ChongGyu Lee [11] independently. His proof uses a global method and is based on the effectiveness of a certain divisor (as done for $N=2$ in [7]).

Corollary C** (cf. Theorem 7.1).**

Let $f:{\mathbb{A}}^{N}\to{\mathbb{A}}^{N}$ be a regular polynomial automorphism over a number field $K$. With the notation as above, there exists a constant $c\geq 0$ such that

[TABLE]

for all $x\in{\mathbb{A}}^{N}(\overline{K})$. Further, we have

[TABLE]

Since (0.2) holds, by the argument of [7], we recover the results on $f$-periodic points and refine the results on non $f$-periodic points in [17, 5, 12, 13]. For $x\in{\mathbb{A}}^{N}(\overline{K})$, let $O_{f}(x):=\{f^{n}(x)\mid n\in{\mathbb{Z}}\}$ denote the $f$-orbit of $x$. If $O_{f}(x)$ is infinite, we have the canonical height $\widehat{h}(O_{f}(x))$ of $O_{f}(x)$ (see Eqn. (7.5)).

Corollary D** (cf. Corollary 7.5, Corollary 7.4).**

Let $f:{\mathbb{A}}^{N}\to{\mathbb{A}}^{N}$ be a regular polynomial automorphism over a number field $K$. With the notation as above, we have the following.

1. (1)

The set of $f$-periodic points in ${\mathbb{A}}^{N}(\overline{K})$ is a set of bounded height. 2. (2)

For any infinite orbit $O_{f}(x)$,

[TABLE]

as $T\to+\infty$.

Acknowledgment. My deep thanks go to Professor Joseph H. Silverman for his encouragement and discussions. Some preliminary work was done while I was staying at the IMJ in Paris and the CRM in Barcelona from October of 2005 to July of 2006. I thank Professors Tien-Cuong Dinh, Charles Favre and Nessim Sibony for helpful conversations, with whom I had opportunities to talk during my stay in Paris.

1. Non-archimedean Green functions for polynomial maps

Let $\Omega$ be an algebraically closed field with non-trivial non-archimedean absolute value $|\cdot|$, and $R$ its ring of integers. For a point $x=(x_{1},\ldots,x_{N})\in{\mathbb{A}}^{N}(\Omega)$, the norm of $x$ is defined by $\|x\|=\max_{i=1,\ldots,N}\{|x_{i}|\}$. As usual, we set $\log^{+}(a):=\log\max\{a,1\}$ for $a\in{\mathbb{R}}_{\geq 0}$, so that $\log^{+}\|x\|=\log\max\{\|x\|,1\}=\log\|(x,1)\|$.

Let $f=(f_{1},\ldots,f_{N}):{\mathbb{A}}^{N}\to{\mathbb{A}}^{N}$ be a polynomial map of degree $d\geq 2$ defined over $\Omega$, where $f_{1}(X),\ldots,f_{N}(X)$ are polynomials in $\Omega[X_{1},\ldots,X_{N}]$ such that $d=\max_{i=1,\ldots,N}\{\deg f_{i}\}$. We write $F_{i}(X,T):=T^{d}f_{i}(X/T)\in\Omega[X_{1},\ldots,X_{N},T]$ for homogenization of $f_{i}$. Let $\overline{f}=(F_{1}:\cdots:F_{N}:T^{d}):{\mathbb{P}}^{N}\dasharrow{\mathbb{P}}^{N}$ denote the extension of $f$ to ${\mathbb{P}}^{N}$. We put $F:=(F_{1},\ldots,F_{N},T^{d}):{\mathbb{A}}^{N+1}\to{\mathbb{A}}^{N+1}$, which is a lift of $\overline{f}$.

For the composition $f^{n}=f\circ\cdots\circ f$, we write $f^{n}=(f_{1}^{n},\ldots,f_{N}^{n})$. Similarly, for the composition $F^{n}=F\circ\cdots\circ F$, we write $F^{n}=(F_{1}^{n},\ldots,F_{N}^{n},T^{d^{n}})$. Let $d_{n}$ denote the degree of $f^{n}$, and let $F_{ni}(X,T)=T^{d_{n}}f_{i}^{n}(X/T)\in\Omega[X_{1},\ldots,X_{N},T]$ be homogenization of $f_{i}^{n}$. Since $F_{i}^{n}(X,1)=f_{i}^{n}(X)=F_{ni}(X,1)$, the degree counting gives $F_{i}^{n}(X,T)=T^{d^{n}-d_{n}}F_{ni}(X,T)$.

Proposition 1.1**.**

Let $f:{\mathbb{A}}^{N}\to{\mathbb{A}}^{N}$ be a polynomial map of degree $d\geq 2$ defined over $\Omega$. Then, for all $x\in{\mathbb{A}}^{N}(\Omega)$, $\displaystyle{\frac{1}{d^{n}}\log^{+}\|f^{n}(x)\|}$ converges to a non-negative real number as $n\to+\infty$.

Proof. We take an $r\in R$ so that $rF_{i}\in R[X,T]$ for all $i=1,\ldots,N$. We set

[TABLE]

where $rF=(rF_{1},\ldots,rF_{N},rT^{d})$. We claim that

[TABLE]

Indeed, the first equality follows from

[TABLE]

The second equality follows from $(rF)^{n}=r^{1+d+\cdots+d^{n-1}}F^{n}=r^{\frac{d^{n}-1}{d-1}}F^{n}$. It follows from $\|(rF)(x,1)\|\leq\|(x,1)\|^{d}$ that

[TABLE]

In other words, $\{c_{n}\}_{n=1}^{+\infty}$ is a non-increasing sequence. Eqn. (1.1) implies that $\{c_{n}\}_{n=1}^{+\infty}$ is bounded from below. Indeed, since $a_{n}$ is nonnegative and $|r|\leq 1$, we have $c_{n}\geq a_{n}+\frac{1}{d-1}\log|r|\geq\frac{1}{d-1}\log|r|$. Thus $\lim_{n\to+\infty}c_{n}$ exists. Eqn. (1.1) then gives the existence of $\lim_{n\to+\infty}a_{n}$, which is nonnegative from the definition.   $\Box$

Proposition 1.1 allows the following definition.

Definition 1.2**.**

For a polynomial map $f:{\mathbb{A}}^{N}\to{\mathbb{A}}^{N}$ defined over $\Omega$, we define the non-negative function $G_{f}:{\mathbb{A}}^{N}(\Omega)\to{\mathbb{R}}$ by

[TABLE]

and call it the Green function for $f$.

Lemma 1.3**.**

Let $C_{f}^{\prime}$ be the maximum of the absolute value of all the coefficients of $f_{i}(X)$ for $1\leq i\leq N$, and we set $c_{f}=\frac{1}{d-1}\log\max\{C_{f}^{\prime},1\}$. Then

[TABLE]

Proof. We take $r\in R$ such that $|r|=\frac{1}{\max\{C_{f}^{\prime},1\}}$. Then $rF_{i}\in R[X,T]$ for all $i=1,\ldots,N$. From the proof of Proposition 1.1, we have

[TABLE]

Hence we get the assertion.   $\Box$

Lemma 1.4 below shows that, for some polynomial maps $f$, $G_{f}$ is not interesting. However, we will see in the next section that $G_{f}$ enjoys nice properties for regular polynomial automorphisms $f$ (see Definition 2.1 and Theorem 2.3).

To state Lemma 1.4, we recall that a polynomial map $f$ is said to be algebraically stable if $d_{n}=d^{n}$ for all $n\geq 1$ ([16, §1.4]).

Lemma 1.4**.**

If $f$ is not algebraically stable, then $G_{f}(x)=0$ for all $x\in{\mathbb{A}}^{N}(\Omega)$.

Proof. We take $n_{0}$ such that $d_{n_{0}}<d^{n_{0}}$, and we put $g=f^{n_{0}}$. Proposition 1.1 tells us that $\frac{1}{d_{n_{0}}^{m}}\log^{+}\|g^{m}(x)\|$ converges to a non-negative number as $m\to+\infty$. Hence

[TABLE]

From Proposition 1.1, we get $G_{f}(x)=0$.   $\Box$

2. Non-archimedean Green functions for regular automorphisms

In this section we consider polynomial automorphisms. Let $f:{\mathbb{A}}^{N}\to{\mathbb{A}}^{N}$ be a polynomial automorphism of degree $d\geq 2$ defined over an algebraically closed field $\Omega$ with non-trivial non-archimedean absolute value.

As before, let $\overline{f}=(F_{1}(X,T):\cdots:F_{N}(X,T):T^{d}):{\mathbb{P}}^{N}\dasharrow{\mathbb{P}}^{N}$ denote the extension of $f$ to ${\mathbb{P}}^{N}$. For the inverse $f^{-1}:{\mathbb{A}}^{N}\to{\mathbb{A}}^{N}$ of $f$, we denote by $d_{-}$ the degree of $f^{-1}$. Note that the integer $d_{-}\geq 2$ may be different from $d$. We write $\overline{f^{-1}}=(G_{1}(X,T):\cdots:G_{N}(X,T):T^{d_{-}}):{\mathbb{P}}^{N}\dasharrow{\mathbb{P}}^{N}$ for the extension of $f^{-1}$ to ${\mathbb{P}}^{N}$.

Let $I_{+}$ and $I_{-}$ denote the set of indeterminacy of $\overline{f}$ and $\overline{f^{-1}}$, respectively:

[TABLE]

Definition 2.1** (see [16, §2.2]).**

A polynomial automorphism $f:{\mathbb{A}}^{N}\to{\mathbb{A}}^{N}$ is called regular if $I_{+}\cap I_{-}=\emptyset$.

Remark 2.2**.**

The definition of regular polynomial automorphisms works over any algebraically closed field.

The purpose of this section is to prove the following theorem, which says that the Green functions for regular automorphisms exhibit nice properties.

Theorem 2.3**.**

Let $\Omega$ be an algebraically closed field with non-trivial non-archimedean valuation, and $f:{\mathbb{A}}^{N}\to{\mathbb{A}}^{N}$ a regular polynomial automorphism over $\Omega$. Then there are open subsets ${V}^{+},{V}^{-}$ of ${\mathbb{A}}^{N}(\Omega)$ and constants $c^{+},c^{-}\in{\mathbb{R}}$ with the following properties.

1. (i)

$G_{f}(\cdot)\geq\log^{+}\|\cdot\|+c^{+}$* on ${V}^{+}$.* 2. (ii)

$G_{f^{-1}}(\cdot)\geq\log^{+}\|\cdot\|+c^{-}$* on ${V}^{-}$.* 3. (iii)

${V}^{+}\cup{V}^{-}={\mathbb{A}}^{N}(\Omega)$.

Remark 2.4**.**

Over ${\mathbb{C}}$, corresponding results (and much more) were established by Sibony [16, §2.2]. Here, since ${\mathbb{A}}^{N}(\Omega)$ is not locally compact in general, we give a different proof that is more algebraic in nature based on Hilbert’s Nullstellensatz. We also give $V^{+},V^{-}$ and $c^{+},c^{-}$ with precise estimates, so that they work well when we introduce the notion of good reduction in §4.

Before proving Theorem 2.3, we show several lemmas. We begin by introducing some notation. Since $I_{+}\cap I_{-}$ is empty, $F_{1}(X,0),\ldots,F_{N}(X,0),G_{1}(X,0),\ldots,G_{N}(X,0)$ have no solutions in common other than [math]. Thus, for each $1\leq i\leq N$, there are polynomials $P_{ij}(X),Q_{ij}(X)\in\Omega[X]$ for $1\leq j\leq N$ such that

[TABLE]

with some $m\geq 1$. Hence there is a polynomial $R_{i}(X,T)\in\Omega[X,T]$ such that

[TABLE]

Here we may and do assume that $m$ is independent of $i$. Replacing $P_{ij}(X)$ by its homogeneous degree $m-d$ part, $Q_{ij}(X)$ by its homogeneous degree $m-d_{-}$ part, and $R_{i}(X,T)$ by its homogeneous degree $m-1$ part, we may and do assume that the $P_{ij}(X)$, $Q_{ij}(X)$ and $R_{i}(X,T)$ are homogeneous polynomials with degree $m-d$, $m-d_{-}$ and $m-1$, respectively.

Let $C^{\prime}$ be the maximum of the absolute value of all the coefficients of $P_{ij}(X)$, $Q_{ij}(X)$ and $R_{i}(X,T)$ for $1\leq i\leq N$ and $1\leq j\leq N$. We set

[TABLE]

We fix real numbers $\varepsilon>0,\delta>0$ satisfying

[TABLE]

Since $C\geq 1$, we have $0<\varepsilon,\delta\leq 1$. For example, we can take

[TABLE]

We define $N_{\delta,\varepsilon}^{+}$ and ${V}_{\delta,\varepsilon}^{+}$ by

[TABLE]

Remark 2.5**.**

We set

[TABLE]

Then $N_{\delta,\varepsilon}^{+}=\overline{N}_{\delta,\varepsilon}^{+}\cap{\mathbb{A}}^{N}(\Omega)$. If $(x:t)\in I_{+}$, then $t=0$ and $F(x,t)=0$. Thus $|t|=0$ and $\|(F(x,t),t^{d})\|=0$, so we have

[TABLE]

Intuitively, points in $N_{\delta,\varepsilon}^{+}$ are near to the hyperplane $\{(x:0)\in{\mathbb{P}}^{N}(\Omega)\}$ at infinity (measured by $\varepsilon$), and also near to $I_{+}$ in “the direction of $x$” (measured by $\delta$).

Lemma 2.6**.**

$f({V}_{\delta,\varepsilon}^{+})\subseteq{V}_{\delta,\varepsilon}^{+}$.

Proof. Taking the complement, it suffices to show that

[TABLE]

Suppose that $x=(x_{1},\ldots,x_{N})\in N_{\delta,\varepsilon}^{+}$. Without loss of generality, we may assume that $|x_{1}|=\|x\|$. We note that $f(x)=(F_{1}(x,1),\ldots,F_{N}(x,1))$ and $f^{-1}(x)=(G_{1}(x,1),\ldots,G_{N}(x,1))$. Since $\varepsilon\leq 1$, we have $\|x\|>1$. Then the definition of $N_{\delta,\varepsilon}^{+}$ gives

[TABLE]

We need to show that $f^{-1}(x)\in N_{\delta,\varepsilon}^{+}$, which is equivalent to

[TABLE]

First we show (2.9). To derive a contradiction, we assume that $\|f^{-1}(x)\|\leq\frac{1}{\varepsilon}$. Let $\lambda>0$ be any small number. We have

[TABLE]

Since $\lambda>0$ is arbitrary, (2.2) and the assumption that $|x_{1}|=\|x\|$ then gives either $\|x\|^{m}\leq C\|x\|^{m-1}$ or $\|x\|^{m}<C\delta\|x\|^{m}$. Equivalently, we have either $\|x\|\leq C$ or $1<C\delta$. However, the former contradicts with (2.4) and (2.7), while the latter contradicts with (2.4). Hence we get (2.9).

Next we show (2.10). Since $\|x\|>\frac{1}{\varepsilon}\geq 1$ from (2.7) and $\|f^{-1}(x)\|>\frac{1}{\varepsilon}\geq 1$ from (2.9), the condition (2.10) is equivalent to $\|x\|<\delta\|f^{-1}(x)\|^{d}$. To derive a contradiction, we assume the contrary, i.e., $\|x\|\geq\delta\|f^{-1}(x)\|^{d}$. Letting $\lambda>0$ be any small number, we have

[TABLE]

Since $\lambda>0$ is arbitrary, (2.2) and the assumption that $|x_{1}|=\|x\|$ gives this time

[TABLE]

However, the former contradicts with (2.4) and (2.7), while the latter contradicts with (2.4). Hence we get (2.10), which completes the proof.   $\Box$

Lemma 2.7**.**

Set $C_{\delta,\varepsilon}^{+}:=\min\{\delta,\varepsilon^{d}\}$. Then

[TABLE]

Proof. For $x\in{V}_{\delta,\varepsilon}^{+}$, the definition of ${V}_{\delta,\varepsilon}^{+}$ gives

[TABLE]

If the latter holds, then we get the assertion since $\delta\geq C_{\delta,\varepsilon}^{+}$. If the former holds, then $C_{\delta,\varepsilon}^{+}\|x\|^{d}\leq 1$. Noting that $C_{\delta,\varepsilon}^{+}\leq 1$, we get $\max\{\|f(x)\|,1\}\geq 1\geq C_{\delta,\varepsilon}^{+}\cdot\max\{\|x\|^{d},1\}$.   $\Box$

Lemma 2.8**.**

Set $c_{\delta,\varepsilon}^{+}:=\frac{1}{d-1}\log C_{\delta,\varepsilon}^{+}$. Then

[TABLE]

Proof. Suppose $x\in{V}_{\delta,\varepsilon}^{+}$. It follows from Lemma 2.6 that $f^{n}(x)\in{V}_{\delta,\varepsilon}^{+}$ for all $n\geq 1$. Then Lemma 2.7 gives

[TABLE]

The usual telescoping argument tells us that

[TABLE]

This completes the proof.   $\Box$

With $f^{-1}$ in place of $f$, we define $N_{\delta,\varepsilon}^{-}$ and ${V}_{\delta,\varepsilon}^{-}$ by

[TABLE]

Then, setting $c_{\delta,\varepsilon}^{-}:=\frac{1}{d_{-}-1}\log\min\{\delta,\epsilon^{d_{-}}\}$, we have

[TABLE]

Lemma 2.9**.**

${V}_{\delta,\varepsilon}^{+}\cup{V}_{\delta,\varepsilon}^{-}={\mathbb{A}}^{N}(\Omega)$.

Proof. Taking the complement, it suffices to show that $N_{\delta,\varepsilon}^{+}\cap N_{\delta,\varepsilon}^{-}=\emptyset$. To derive a contradiction, we assume that there is an $x\in N_{\delta,\varepsilon}^{+}\cap N_{\delta,\varepsilon}^{-}$. Then we have

[TABLE]

Without loss of generality, we assume that $|x_{1}|=\|x\|$. Let $\lambda>0$ be any small number. By (2.13)–(2.15), we have

[TABLE]

Since $\lambda$ is arbitrary, it follows from (2.2) that either $\|x\|^{m}<C\delta\|x\|^{m}$ or $\|x\|^{m}\leq C\|x\|^{m-1}$. Hence we get

[TABLE]

However, the former contradicts with (2.4), while the latter contradicts with (2.4) and (2.13). Thus we have $N_{\delta,\varepsilon}^{+}\cap N_{\delta,\varepsilon}^{-}=\emptyset$.   $\Box$

Proof of Theorem 2.3. Let $\delta$ and $\varepsilon$ be constants satisfying (2.4). Then Theorem 2.3 holds with ${V}^{\pm}={V}_{\delta,\varepsilon}^{\pm}$ and $c^{\pm}=c_{\delta,\varepsilon}^{\pm}$. Indeed, the condition (i) follows from Lemma 2.8, and the condition (ii) from (2.12), while the condition (iii) follows from Lemma 2.9.   $\Box$

3. Non-archimedean Green functions and the set of

escaping points

In this section, we continue to study some basic properties of regular polynomial automorphisms defined over $\Omega$. We keep the notation and the assumption of §2. In particular, $f:{\mathbb{A}}^{N}\to{\mathbb{A}}^{N}$ denotes a regular polynomial automorphism of degree $d\geq 2$ defined over $\Omega$.

In analogy with the field of complex numbers, we define the set $W^{+}$ of escaping points and the set ${\mathcal{K}}^{+}$ of non-escaping points by

[TABLE]

Then the following theorem holds, which is a non-archimedean version of the results of [1, §§2–3] and [16, §2].

Theorem 3.1**.**

Let $f:{\mathbb{A}}^{N}\to{\mathbb{A}}^{N}$ be a regular polynomial automorphism over $\Omega$, and let $G_{f}$ be the Green function for $f$.

1. (1)

The set ${\mathcal{K}}^{+}$ is exactly the set of points where $G_{f}$ vanish:**

[TABLE] 2. (2)

${\mathbb{A}}^{N}(\Omega)=W^{+}\amalg{\mathcal{K}}^{+}$* (disjoint union).*

To prove Theorem 3.1, we need the following two lemmas. Recall that $\delta$ and $\varepsilon$ are fixed constants satisfying (2.4).

Lemma 3.2**.**

For any $x\in N_{\delta,\frac{\varepsilon}{2}}^{+}$, one has $\|x\|\leq\frac{1}{2}\|f^{-1}(x)\|$.

Proof. It follows from $x\in N_{\delta,\frac{\varepsilon}{2}}^{+}$ that

[TABLE]

To derive a contradiction, we assume that $\|x\|>\frac{1}{2}\|f^{-1}(x)\|$. Without loss of generality, we assume that $|x_{1}|=\|x\|$. Then (we take $\lambda=C$ here)

[TABLE]

Using (2.2), we get

[TABLE]

However, the former contradicts with (2.4). If the latter holds, then Eqn. (3.1) implies $1<C\varepsilon$, contradicting with (2.4). This completes the proof.   $\Box$

Lemma 3.3**.**

For any $x\in{\mathbb{A}}^{N}(\Omega)$, one has $f^{n}(x)\in V_{\delta,\frac{\varepsilon}{2}}^{+}$ for all sufficiently large $n$.

Proof. Note that $\frac{\varepsilon}{2},\delta$ satisfies (2.4) with $\frac{\varepsilon}{2}$ in place of $\varepsilon$. Thus, if $x\in V_{\delta,\frac{\varepsilon}{2}}^{+}$, then Lemma 2.6 gives $f^{n}(x)\in V_{\delta,\frac{\varepsilon}{2}}^{+}$ for all $n\geq 0$.

Suppose now that $x\in N_{\delta,\frac{\varepsilon}{2}}^{+}$. We take a positive integer $n_{0}$ so that $\|x\|\leq\frac{2^{n_{0}+1}}{\varepsilon}$. We claim that $f^{n_{0}}(x)\in V_{\delta,\frac{\varepsilon}{2}}^{+}$. Indeed, if we assume the contrary, then Lemma 3.2, applied to $x,\ldots,f^{n_{0}}(x)\in N_{\delta,\frac{\varepsilon}{2}}^{+}$, gives

[TABLE]

which contradicts with our choice of $n_{0}$. Thus $f^{n}(x)\in V_{\delta,\frac{\varepsilon}{2}}^{+}$ for all $n\geq n_{0}$.   $\Box$

Proof of Theorem 3.1. (1) From Definition 1.2, we get ${\mathcal{K}}^{+}\subseteq\{x\in{\mathbb{A}}^{N}(\Omega)\mid G_{f}(x)=0\}$. To show the other inclusion, we assume that $G_{f}(x)=0$. Then $G_{f}(f^{n}(x))=d^{n}G_{f}(x)=0$ for all $n\geq 0$. By Lemma 3.3, we take $n_{0}$ such that $f^{n_{0}}(x)\in V_{\delta,\frac{\varepsilon}{2}}^{+}$. It follows from Lemma 2.6 and Lemma 2.8 (applied to $\frac{\varepsilon}{2}$ in place of $\varepsilon$) that

[TABLE]

for all $n\geq n_{0}$. Combined with $G_{f}(f^{n}(x))=0$, we see that $\|f^{n}(x)\|\leq\exp\left(-c_{\delta,\frac{\varepsilon}{2}}^{+}\right)$ for all $n\geq n_{0}$. Thus $\{x\in{\mathbb{A}}^{N}(\Omega)\mid G_{f}(x)=0\}\subseteq{\mathcal{K}}^{+}$.

(2) If $x\not\in{\mathcal{K}}^{+}$, then $G_{f}(x)>0$ by (1). Definition 1.2 then gives $\|f^{n}(x)\|\to+\infty$ as $n\to+\infty$.   $\Box$

With $f^{-1}$ in place of $f$, we put

[TABLE]

Then we have ${\mathbb{A}}^{N}(\Omega)=W^{-}\amalg{\mathcal{K}}^{-}$ as in Theorem 3.1.

In the rest of this section, we give filtrations of ${\mathbb{A}}^{N}$ relative to $f$ over non-archimedean fields, as in Bedford–Smillie [1, §2.2] and [15, §3] over ${\mathbb{C}}$.

We set

[TABLE]

where $\delta$ and $\varepsilon$ are constants satisfying (2.4).

Since $\varepsilon\leq 1$ and $\frac{\delta}{\varepsilon^{d}}\geq C^{d}\geq 1$ by (2.4), we have

[TABLE]

so that $B_{\delta,\varepsilon}\amalg U^{+}_{\delta,\varepsilon}=V^{+}_{\delta,\varepsilon}$.

Proposition 3.4**.**

We assume that $\varepsilon$ and $\delta$ satisfy

[TABLE]

in addition to $\eqref{eqn:e:d}$ (For example, if we take $\varepsilon$ and $\delta$ as (2.5), then they also satisfy (3.2)). Then we have the following.

1. (1)

${\mathbb{A}}^{N}(\Omega)=B_{\delta,\varepsilon}\amalg U^{+}_{\delta,\varepsilon}\amalg U^{-}_{\delta,\varepsilon}$* (disjoint union).* 2. (2)

$f(U^{+}_{\delta,\varepsilon})\subseteq U^{+}_{\delta,\varepsilon}$* and $f(B_{\delta,\varepsilon}\amalg U^{+}_{\delta,\varepsilon})\subseteq B_{\delta,\varepsilon}\amalg U^{+}_{\delta,\varepsilon}$.* 3. (3)

$f^{-1}(U^{-}_{\delta,\varepsilon})\subseteq U^{-}_{\delta,\varepsilon}$* and $f^{-1}(B_{\delta,\varepsilon}\amalg U^{-}_{\delta,\varepsilon})\subseteq B_{\delta,\varepsilon}\amalg U^{-}_{\delta,\varepsilon}$.*

Proof. (1) This is obvious from the definition.

(2) Since $B_{\delta,\varepsilon}\amalg U^{+}_{\delta,\varepsilon}=V^{+}_{\delta,\varepsilon}$, we have $f(B_{\delta,\varepsilon}\amalg U^{+}_{\delta,\varepsilon})\subseteq B_{\delta,\varepsilon}\amalg U^{+}_{\delta,\varepsilon}$ by Lemma 2.6.

Suppose that $x\in U^{+}_{\delta,\varepsilon}$. Then

[TABLE]

where we have used (3.2) in the last inequality. Also, since $x\in U^{+}_{\delta,\varepsilon}\subseteq V^{+}_{\delta,\varepsilon}$, we have $f(x)\in V^{+}_{\delta,\varepsilon}$ by Lemma 2.6. Since $f(x)\not\in B_{\delta,\varepsilon}$ by (3.3), we get $f(x)\in V^{+}_{\delta,\varepsilon}\setminus B_{\delta,\varepsilon}=U^{+}_{\delta,\varepsilon}$. Hence $f(U^{+}_{\delta,\varepsilon})\subseteq U^{+}_{\delta,\varepsilon}$.

(3) We put

[TABLE]

where the second equality follows from $\frac{\delta}{\varepsilon^{d}_{-}}\geq C^{d_{-}}\geq 1$ by (2.4). Then as in (2), we have $f^{-1}(\widetilde{U}^{-}_{\delta,\varepsilon})\subseteq\widetilde{U}^{-}_{\delta,\varepsilon}$. Since $B_{\delta,\varepsilon}\amalg\widetilde{U}^{-}_{\delta,\varepsilon}=V^{-}_{\delta,\varepsilon}$, Lemma 2.9 implies that $U^{-}_{\delta,\varepsilon}\subseteq\widetilde{U}^{-}_{\delta,\varepsilon}$.

Suppose that $x\in U^{-}_{\delta,\varepsilon}$. Then

[TABLE]

In particular, $\|f^{-1}(x)\|>\frac{1}{\varepsilon}$, so that $f^{-1}(x)\not\in B_{\delta,\varepsilon}$. On the other hand, since $x\not\in U^{+}_{\delta,\varepsilon}$ and $f(U^{+}_{\delta,\varepsilon})\subseteq U^{+}_{\delta,\varepsilon}$, we get $f^{-1}(x)\not\in U^{+}_{\delta,\varepsilon}$. It follows that $f^{-1}(x)\in U^{-}_{\delta,\varepsilon}={\mathbb{A}}^{N}(\Omega)\setminus(B_{\delta,\varepsilon}\amalg U^{+}_{\delta,\varepsilon})$. We conclude that $f^{-1}(U^{-}_{\delta,\varepsilon})\subseteq U^{-}_{\delta,\varepsilon}$.

Next we show $f^{-1}(B_{\delta,\varepsilon}\amalg U^{-}_{\delta,\varepsilon})\subseteq B_{\delta,\varepsilon}\amalg U^{-}_{\delta,\varepsilon}$. Since $U^{+}_{\delta,\varepsilon}={\mathbb{A}}^{N}(\Omega)\setminus(B_{\delta,\varepsilon}\amalg U^{-}_{\delta,\varepsilon})$, it suffices to show that $f^{-1}(U^{+}_{\delta,\varepsilon})\supseteq U^{+}_{\delta,\varepsilon}$. This inclusion is obvious from $f(U^{+}_{\delta,\varepsilon})\subseteq U^{+}_{\delta,\varepsilon}$.   $\Box$

Proposition 3.5**.**

We assume that $\varepsilon$ and $\delta$ satisfy

[TABLE]

in addition to $\eqref{eqn:e:d}$. Then we have the following.

1. (1)

$\bigcup_{n=0}^{+\infty}f^{-n}(U^{+}_{\delta,\varepsilon})=W^{+}$. 2. (2)

$\bigcup_{n=0}^{+\infty}f^{n}(U^{-}_{\delta,\varepsilon})=W^{-}$.

Proof. (1) We set $r:=\frac{\delta}{\varepsilon^{d-1}}>1$. We first show that $U^{+}_{\delta,\varepsilon}\subseteq W^{+}$. Indeed, if $x\in U^{+}_{\delta,\varepsilon}$, then

[TABLE]

Since $f(U^{+}_{\delta,\varepsilon})\subseteq U^{+}_{\delta,\varepsilon}$, we inductively get $\|f^{n}(x)\|>r^{\frac{d^{n}-1}{d-1}}\frac{1}{\varepsilon}$ for all $n\geq 0$. Hence $x\in W^{+}$. This completes the proof of $U^{+}_{\delta,\varepsilon}\subseteq W^{+}$. Since $f^{-1}(W^{+})=W^{+}$, we get $f^{-n}(U^{+}_{\delta,\varepsilon})\subseteq W^{+}$ for all $n\geq 0$, so that $\bigcup_{n=0}^{+\infty}f^{-n}(U^{+}_{\delta,\varepsilon})\subseteq W^{+}$.

To show the other inclusion $\bigcup_{n=0}^{+\infty}f^{-n}(U^{+}_{\delta,\varepsilon})\supseteq W^{+}$, suppose that $x\not\in\bigcup_{n=0}^{+\infty}f^{-n}(U^{+}_{\delta,\varepsilon})$. We need to show that $x\in{\mathcal{K}}^{+}$. Since $f^{n}(x)\not\in U^{+}_{\delta,\varepsilon}$, we have either $f^{n}(x)\in B_{\delta,\varepsilon}$ or $f^{n}(x)\in U^{-}_{\delta,\varepsilon}$.

Case 1: Suppose that there is an $n_{0}\geq 0$ such that $f^{n_{0}}(x)\in B_{\delta,\varepsilon}$. Then $f^{n_{0}+1}(x)\in B_{\delta,\varepsilon}\amalg U^{+}_{\delta,\varepsilon}$ by Proposition 3.4(2). Since $f^{n_{0}+1}(x)\not\in U^{+}_{\delta,\varepsilon}$, we obtain $f^{n_{0}+1}(x)\in B_{\delta,\varepsilon}$. Inductively, $f^{n}(x)\in B_{\delta,\varepsilon}$ for all $n\geq n_{0}$, so we conclude that $x\in{\mathcal{K}}^{+}$.

Case 2: Suppose that $f^{n}(x)\in U^{-}_{\delta,\varepsilon}$ for all $n\geq 0$. By Lemma 3.3, there is an $n_{0}\geq 0$ such that $f^{n}(x)\in V_{\delta,\frac{\varepsilon}{2}}^{+}$ for all $n\geq n_{0}$. Then for all $n\geq n_{0}$, we have

[TABLE]

Hence $x\in{\mathcal{K}}^{+}$.

In both cases we have $x\in{\mathcal{K}}^{+}$, so we get $\bigcup_{n=0}^{+\infty}f^{-n}(U^{+}_{\delta,\varepsilon})\supseteq W^{+}$.

(2) Let $\widetilde{U}^{-}_{\delta,\varepsilon}$ be the set defined by (3.4). Then the argument in (1) gives $\bigcup_{n=0}^{+\infty}f^{n}(\widetilde{U}^{-}_{\delta,\varepsilon})=W^{-}$, and so $\bigcup_{n=0}^{+\infty}f^{n}(U^{-}_{\delta,\varepsilon})\subseteq W^{-}$. To show the other inclusion, suppose that $x\not\in\bigcup_{n=0}^{+\infty}f^{n}(U^{-}_{\delta,\varepsilon})$. Then we have either $f^{-n}(x)\in B_{\delta,\varepsilon}$ or $f^{-n}(x)\in U^{+}_{\delta,\varepsilon}$.

Case 1: If there is an $n_{0}\geq 0$ such that $f^{-n_{0}}(x)\in B_{\delta,\varepsilon}$, then the argument of Case 1 of (1), together with Proposition 3.4(3), gives $f^{-n}(x)\in B_{\delta,\varepsilon}$ for all $n\geq n_{0}$.

Case 2: Suppose that $f^{-n}(x)\in U^{+}_{\delta,\varepsilon}$ for all $n\geq 0$.

then the argument of Case 2 of (4), together with Lemma 3.3 with $f^{-1}$ in place of $f$, gives $\frac{1}{\varepsilon}<\|x\|<\frac{2}{\varepsilon}$ for sufficiently large $n$.

In both cases, we get $x\in{\mathcal{K}}^{-}$. Hence $\bigcup_{n=0}^{+\infty}f^{n}(U^{-}_{\delta,\varepsilon})\supseteq W^{-}$.   $\Box$

Remark 3.6**.**

If we take

[TABLE]

then they satisfy both (2.4) and (3.5).

4. Regular automorphisms having good reduction

In [14], Morton and Silverman introduced the notion of having good reduction for endomorphisms of ${\mathbb{P}}^{1}$ over $\Omega$, which has been useful in studying endomorphisms of ${\mathbb{P}}^{1}$ over a global field. For endomorphisms of ${\mathbb{P}}^{N}$ having good reduction, see, for example [8, Remark 12] and [9]. In this section, we introduce the notion of having good reduction for regular polynomial automorphisms of ${\mathbb{A}}^{N}$ over $\Omega$. This notion will be useful in studying regular polynomial automorphisms over a global field in §§6–7.

As in §1, $R$ denotes the ring of integers of $\Omega$. Let $M$ be the maximal ideal of $R$, and $\widetilde{k}:=R/M$ the residue field. Note that $\widetilde{k}$ is algebraically closed since $\Omega$ is algebraically closed.

Definition 4.1** (good reduction).**

Let $f=(f_{1},\ldots,f_{N}):{\mathbb{A}}^{N}\to{\mathbb{A}}^{N}$ be a regular polynomial automorphism over an algebraically closed field $\Omega$ with non-trivial non-archimedean absolute value, and let $f^{-1}=(g_{1},\ldots,g_{N}):{\mathbb{A}}^{N}\to{\mathbb{A}}^{N}$ denote its inverse. We write $d$ and $d_{-}$ for the degrees of $f$ and $f^{-1}$, respectively. We say that $f$ has good reduction if the following three conditions are satisfied.

1. (i)

$f$ extends to the polynomial automorphism $f:{\mathbb{A}}^{N}_{R}\to{\mathbb{A}}^{N}_{R}$ over $R$, meaning that both $f_{1}(X),\ldots,f_{N}(X)$ and $g_{1}(X),\ldots,g_{N}(X)$ are in $R[X_{1},\ldots,X_{N}]$. 2. (ii)

Let $\widetilde{f}=(\widetilde{f_{1}},\ldots,\widetilde{f_{N}}):{\mathbb{A}}^{N}_{\widetilde{k}}\to{\mathbb{A}}^{N}_{\widetilde{k}}$ and $\widetilde{f^{-1}}=(\widetilde{g_{1}},\ldots,\widetilde{g_{N}}):{\mathbb{A}}^{N}_{\widetilde{k}}\to{\mathbb{A}}^{N}_{\widetilde{k}}$ be the induced polynomial automorphisms over $\widetilde{k}$. Then the degrees of $\widetilde{f}$ and $\widetilde{f^{-1}}$ are equal to $d$ and $d_{-}$, respectively. 3. (iii)

$\widetilde{f}$ is regular (cf. Remark 2.2).

We give some equivalent conditions for regular polynomial automorphisms $f$ to have good reduction. As in §1, let $F_{i}(X,T)$ and $G_{j}(X,T)$ be the homogenization of $f_{i}(X)$ and $g_{j}(X)$. If $F_{i}(X,T)$ and $G_{j}(X,T)$ are defined over $R$, we denote by $\widetilde{F_{i}}(X,T)$ and $\widetilde{G_{j}}(X,T)$ their reductions to $\widetilde{k}$. Let $\rho:R\to\widetilde{k}$ be the natural map.

Proposition 4.2**.**

Let $f$ be a regular polynomial automorphism of ${\mathbb{A}}^{N}$ over $\Omega$. Assume that $f$ satisfies the conditions (i) and (ii) of Definition 4.1. Then the followings are equivalent.

1. (1)

$f$* has good reduction, i.e., $f$ also satisfies the condition (iii) of Definition 4.1.* 2. (2)

As ideals in $R[X_{1},\ldots,X_{N},T]$, one has

[TABLE]

for some integer $k\geq 1$. 3. (3)

As ideals in $R[X_{1},\ldots,X_{N}]$, one has

[TABLE]

for some integer $\ell\geq 1$.

Proof. (1) $\Longrightarrow$ (3): It suffices to show that

[TABLE]

for some $\ell\geq 1$. We set

[TABLE]

Since $f$ is regular, $I$ is a non-zero ideal of $R$.

We claim that $\rho(I)\neq 0$. Indeed, suppose that $\rho(I)=0$. Then the elimination theory (cf. [8, Theorem 6]) tells us that there is a point $x=(x_{1}:\ldots:x_{n})\in{\mathbb{P}}^{N-1}(\widetilde{k})$ such that $\widetilde{F_{i}}(x,0)=0$ and $\widetilde{G_{j}}(x,0)=0$ for all $i$ and $j$ . Since $f$ satisfies the condition (ii), $\widetilde{F_{i}}(X,T)$ and $\widetilde{G_{j}}(X,T)$ are homogenization of $\widetilde{f_{i}}$ and $\widetilde{g_{j}}$, respectively. Then the existence of such an $x\in{\mathbb{P}}^{N-1}(\widetilde{k})$ contradicts with the condition (iii). Hence we get the claim.

Since $\rho(I)\neq 0$, there is an $r\in I$ such that $r\in R^{\times}=R\setminus M$. Then $I=R$, and we obtain Eqn. (4.1).

(3) $\Longrightarrow$ (1): The assumption of (3) gives, as ideals in $\widetilde{k}[X]$,

[TABLE]

Since $\rho({F_{i}}(X,0))=\widetilde{F_{i}}(X,0)$ and $\rho({G_{j}}(X,0))=\widetilde{G_{j}}(X,0)$, we obtain that $\widetilde{f}$ is regular.

(2) $\Longrightarrow$ (3): We have only to put $T=0$.

(3) $\Longrightarrow$ (2): It suffices to show that, for any $\alpha=1,\ldots,N$, there are an integer $k\geq 1$ and polynomials $P_{i}(X,T)$, $Q_{j}(X,T)$ and $R(X,T)$ defined over $R$ such that

[TABLE]

By the assumption of (iii), there are an integer $\ell\geq 1$ and polynomials $P_{i}(X)$, $Q_{j}(X)$ defined over $R$ such that

[TABLE]

We set $k:=\ell$, $P_{i}(X,T):=P_{i}(X)$ and $Q_{j}(X,T):=Q_{j}(X)$. Then

[TABLE]

is a polynomial in $R[X,T]$, which is divided by $T$. Hence there is a polynomial $R(X,T)$ in $R[X,T]$ satisfying Eqn. (4.2).   $\Box$

Suppose now that a regular polynomial automorphism $f$ has good reduction. By Proposition 4.2, for each $1\leq i\leq N$, there are polynomials $P_{ij}(X)$ and $Q_{ij}(X)$ in $R[X]$ that satisfy (2.1). Then the polynomial $R_{i}(X,T)$ in (2.2) is also defined over $R$. Then the constant $C$ in (2.3) is equal to $1$. This means that $\varepsilon=1$ and $\delta=1$ satisfy (2.4). It follows that, when $f$ has good reduction, $G_{f}$ and $\log^{+}\|\cdot\|$ are related simply.

Proposition 4.3**.**

Suppose that $f$ has good reduction. Then

1. (1)

$G_{f}(\cdot)\leq\log^{+}\|\cdot\|$* and $G_{f^{-1}}(\cdot)\leq\log^{+}\|\cdot\|$ on ${\mathbb{A}}^{N}(\Omega)$.* 2. (2)

$\log^{+}\|\cdot\|\leq G_{f}(\cdot)$* on ${V}_{1,1}^{+}$, $\log^{+}\|\cdot\|\leq G_{f^{-1}}(\cdot)$ on ${V}_{1,1}^{-}$, and ${\mathbb{A}}^{N}(\Omega)={V}_{1,1}^{+}\cup{V}_{1,1}^{-}$.*

Proof. (1) Since the $f_{i}(X)$ are defined over $R$, in the proof of Lemma 1.3 we may take $r=1$, so that $c_{f}=1$. Then $G_{f}(\cdot)\leq\log^{+}\|\cdot\|$ on ${\mathbb{A}}^{N}(\Omega)$. The estimate for $G_{f^{-1}}$ is similar.

(2) Since $\varepsilon=1$ and $\delta=1$ satisfy (2.4), Lemma 2.9 gives ${\mathbb{A}}^{N}(\Omega)={V}_{1,1}^{+}\cup{V}_{1,1}^{-}$. The constant $c^{+}_{1,1}$ in Lemma 2.8 is equal to [math], so we have $\log^{+}\|x\|\leq G_{f}(x)$ for all $x\in{V}_{1,1}^{+}$. The estimate for $G_{f^{-1}}$ is similar.   $\Box$

5. Green functions for regular automorphisms over ${\mathbb{C}}$

In this section, we remark that the proof of Theorem 2.3 gives a different proof (more explicit and without compactness arguments) of the corresponding estimates of Green functions over ${\mathbb{C}}$

We write the usual absolute value of ${\mathbb{C}}$ for $|\cdot|_{\infty}$, and we set $\|x\|_{\infty}:=\max_{i}\{|x_{i}|_{\infty}\}$ for $x=(x_{1},\ldots,x_{N})\in{\mathbb{A}}^{N}({\mathbb{C}})$.

Let $f=(f_{1},\ldots,f_{N}):{\mathbb{A}}^{N}\to{\mathbb{A}}^{N}$ be a regular polynomial automorphism of degree $d\geq 2$ defined over ${\mathbb{C}}$. Then the Green function for $f$ is defined by (see [16, §2])

[TABLE]

Let $\|f\|_{\infty}$ be the maximum of the absolute values of all the coefficients of $f_{i}(X)$ for $1\leq i\leq N$, and we set $c_{f,\infty}=\frac{1}{d-1}\log\max\left\{\binom{N+d-1}{d}\|f\|_{\infty},1\right\}$. Note that $\binom{N+d-1}{d}$ is the number of monomials of degree $d$ in the ring of homogeneous polynomials in $N$ variables. Since

[TABLE]

we get

[TABLE]

Let $P_{ij}(X),Q_{ij}(X)\in{\mathbb{C}}[X]$ and $R(X,T)\in{\mathbb{C}}[X,T]$ be polynomials satisfying (2.2). As before, we may and do assume that the $P_{ij}(X)$, $Q_{ij}(X)$ and $R_{i}(X,T)$ are homogeneous polynomials with degree $m-d$, $m-d_{-}$ and $m-1$, respectively. We write $\|P\|_{\infty}$ for the maximum of the absolute values of all the coefficients of $P_{ij}(X)$ for $1\leq i\leq N$ and $1\leq j\leq N$, and we write $\|Q\|_{\infty}$, $\|R\|_{\infty}$ similarly. We set

[TABLE]

We put

[TABLE]

We fix real numbers $\varepsilon>0,\delta>0$ satisfying (2.4) with $C_{\infty}$ in place of $C$. We define $N^{\pm}_{\delta,\varepsilon}$ and $V^{\pm}_{\delta,\varepsilon}$ by (2.6) and (2.11) with ${\mathbb{C}}$ in place of $\Omega$. Then, exactly as in Theorem 2.3, we have the following.

Theorem 5.1**.**

$f:{\mathbb{A}}^{N}\to{\mathbb{A}}^{N}$* a regular polynomial automorphism over ${\mathbb{C}}$. Then*

1. (i)

$G_{f}(\cdot)\geq\log^{+}\|\cdot\|+c_{\delta,\varepsilon}^{+}$* on $V^{+}_{\delta,\varepsilon}$.* 2. (ii)

$G_{f^{-1}}(\cdot)\geq\log^{+}\|\cdot\|+c_{\delta,\varepsilon}^{-}$* on $V^{-}_{\delta,\varepsilon}$.* 3. (iii)

$V^{+}_{\delta,\varepsilon}\cup V^{-}_{\delta,\varepsilon}={\mathbb{A}}^{N}(\Omega)$.

6. Global theory of regular automorphisms

From this section, we turn our attention to regular automorphisms over a number field.

Let $K$ be a number field, and $O_{K}$ its ring of integers. We fix an embedding $K\subset\overline{K}$ into an algebraic closure. Let $M_{K}$ be the set of absolute values on $K$. We extend the absolute values on $K$ to those on $\overline{K}$.

Let $L$ be a finite extension field of $K$. For $x\in{\mathbb{A}}^{N}(L)$, we define

[TABLE]

where $n_{v}=\frac{[L_{v}:K_{v}]}{[L:K]}$. This gives rise to the logarithmic Weil height function

[TABLE]

For more details on height functions, we refer the reader to [2, 6, 10].

Let $f:{\mathbb{A}}^{N}\to{\mathbb{A}}^{N}$ be a regular polynomial automorphism over $\overline{K}$ (cf. Remark 2.2). If the coefficients of $f$ are all defined over $K$, then we say that $f$ is a regular polynomial automorphism over $K$.

Lemma 6.1**.**

If $f:{\mathbb{A}}^{N}\to{\mathbb{A}}^{N}$ is a regular polynomial automorphism over $K$, then the coefficients of $f^{-1}$ are also all defined over $K$.

Proof. We take a finite Galois extension field $L$ of $K$ such that the coefficients of $f^{-1}$ are elements of $L$. For every $\sigma\in\operatorname{Gal}(L/K)$, the uniqueness of the inverse gives $(f^{-1})^{\sigma}=f^{-1}$. Thus the coefficients of $f^{-1}$ are in fact elements of $K$.   $\Box$

In [7], we constructed (global) canonical height functions $h^{+}_{f}$ and $h^{-}_{f}$ for polynomial automorphisms $f$ over $K$, under the assumption that there exists a constant $c\geq 0$ such that

[TABLE]

for all $x\in{\mathbb{A}}^{N}(\overline{K})$, where $d$ and $d_{-}$ denote the degrees of $f$ and $f^{-1}$. (We showed in op.cit that (6.2) holds for regular polynomial automorphisms in dimension $N=2$ by a global method, i.e., a method using the effectiveness of a certain divisor on a certain rational surface.)

In the following, using properties of local Green functions studied in the previous sections, we will first construct in Theorem 6.3 (global) canonical height functions $h^{+}_{f}$ and $h^{-}_{f}$ for regular polynomial automorphisms. Indeed, we will construct $h^{+}_{f}$ and $h^{-}_{f}$ as appropriate sums of local Green functions. Then, we show local versions of (6.2) for all places $v$, and summing them up we will obtain (6.2) for regular polynomial automorphisms in any dimension $N\geq 2$ in Theorem 7.1.

For a finite set $S$ of $M_{K}$ that contains all the Archimedean absolute values of $K$, we denote by $(O_{K})_{S}$ the ring of $S$-integers:

[TABLE]

Proposition 6.2**.**

Let $f:{\mathbb{A}}^{N}\to{\mathbb{A}}^{N}$ be a regular polynomial automorphism of degree $d\geq 2$ over a number field $K$. Then there exists a finite set $S$ of $M_{K}$ that contains all the Archimedean absolute values of $K$ with the following property:* For all $v\not\in S$, $f$ induces the regular polynomial automorphism over $\overline{K_{v}}$ that has good reduction.*

Proof. We write $f=(f_{1},\ldots,f_{N})$ and let $F_{i}(X,T)\in K[X,T]$ be the homogenization of $f_{i}$. Let $d_{-}$ denote the degree of $f^{-1}=(g_{1},\ldots,g_{N})$, and in virtue of Lemma 6.1 let $G_{j}(X,T)\in K[X,T]$ be the homogenization of $g_{j}$. Then there are an integer $m$ and homogeneous polynomials $P_{ij}(X)\in K[X]$ of degree $m-d$, $Q_{ij}(X)\in K[X]$ of degree $m-d_{-}$, and $R_{i}(X,T)\in K[X,T]$ of degree $m-1$ such that (2.2) holds as polynomials in $K[X,T]$.

We take a a finite set $S$ of $M_{K}$ that contains all the Archimedean absolute values of $K$ with the following properties:

1. (i)

The coefficients of $F_{i}(X,T),G_{j}(X,T),P_{ij}(X),Q_{ij}(X),R_{i}(X,T)$ are all in $(O_{K})_{S}$. 2. (ii)

For $v\not\in S$, we denote by $\rho_{v}:(O_{K})_{S}\to\widetilde{k_{v}}$ the natural map, where $\widetilde{k_{v}}$ is the residue field of $(O_{K})_{v}$. Then $\deg(f)=\deg(\rho_{v}(f))$ and $\deg(f^{-1})=\deg(\rho_{v}(f^{-1}))$.

Then for any $v\not\in S$, $f\times_{K}\overline{K_{v}}:{\mathbb{A}}^{N}_{\overline{K_{v}}}\to{\mathbb{A}}^{N}_{\overline{K_{v}}}$ satisfies the properties (i) and (ii) of Definition 4.1 and (iii) of Proposition 4.2. Hence $f\times_{K}\overline{K_{v}}$ has good reduction.   $\Box$

Theorem 6.3**.**

Let $f:{\mathbb{A}}^{N}\to{\mathbb{A}}^{N}$ be a regular polynomial automorphism of degree $d\geq 2$ over a number field $K$. Let $d_{-}\geq 2$ denote the degree of $f^{-1}$.

1. (1)

For all $x\in{\mathbb{A}}^{N}(\overline{K})$, the limits

[TABLE]

exist. We write $\widehat{h}^{+}_{f}(x)$ and $\widehat{h}^{-}_{f}(x)$ for the limits, respectively. 2. (2)

(Global-to-local decomposition)* For each place $v\in M_{K}$, let $G_{f,v}$ and $G_{f^{-1},v}$ be the Green functions for $f$ and $f^{-1}$ at $v$, respectively. Then, for all $x\in{\mathbb{A}}^{N}(\overline{K})$, one has*

[TABLE] 3. (3)

We define $\widehat{h}_{f}:{\mathbb{A}}^{N}(\overline{K})\to{\mathbb{R}}$ by

[TABLE]

Then $\widehat{h}_{f}$ satisfies the following two conditions.

1. (i)

$\displaystyle\frac{1}{d}\widehat{h}_{f}\circ f+\frac{1}{d_{-}}\widehat{h}_{f}\circ f^{-1}=\left(1+\frac{1}{dd_{-}}\right)\widehat{h}_{f}$* on ${\mathbb{A}}^{N}(\overline{K})$.* 2. (ii)

$\widehat{h}_{f}\gg\ll h$* on ${\mathbb{A}}^{N}(\overline{K})$.* 4. (4)

The function $\widehat{h}_{f}$ has the following uniqueness property:* If $h^{\prime}:{\mathbb{A}}^{N}(\overline{K})\to{\mathbb{R}}$ is a function satisfying the condition (3-i) such that $h^{\prime}=\widehat{h}_{f}+O(1)$, then $h^{\prime}=\widehat{h}_{f}$.* 5. (5)

The functions $\widehat{h}^{+}_{f}$, $\widehat{h}^{-}_{f}$ and $\widehat{h}_{f}$ are non-negative. Further, for $x\in{\mathbb{A}}^{N}(\overline{K})$, we have

[TABLE]

Proof. For each $v\in M_{K}$, we have estimates of Green functions for $f$ at $v$ as in Lemma 1.3 and Lemma 2.8. We use the suffix $v$ when we work over the absolute value $v\in M_{K}$. For example, Green function for $f$ at $v$ is denoted by $G_{f,v}$ and constants $c_{f},c^{\pm}_{\varepsilon,\delta}$ in Lemma 1.3, Lemma 2.8 and (2.12) are denoted by $c_{f,v},c^{\pm}_{\varepsilon,\delta,v}$ respectively.

Let $S$ be a finite subset of $M_{K}$ in Proposition 6.2.

(1)(2) We fix $x\in{\mathbb{A}}^{N}(\overline{K})$. We will show the existence of $h_{f}^{+}(x)$ and the decomposition $h_{f}^{+}(x)=\sum_{v\in M_{K}}n_{v}G_{f,v}(x)$. The existence and decomposition for $h_{f}^{-}(x)$ are shown similarly.

For $v\in M_{K}$ and $n\geq 0$, we set

[TABLE]

Then we have

• •

$0\leq G^{+}_{v,n}(x)\leq\log^{+}\|x\|_{v}+c_{f,v}$ for all $v\in M_{K}$ and $n\geq 0$ from Lemma 1.3 (or its proof) and Eqn. (5.4).

• •

$\lim_{n\to+\infty}G^{+}_{v,n}(x)=G_{f,v}(x)$ from Definition 1.2 and Eqn. (5.3).

• •

$\frac{1}{d^{n}}h(f^{n}(x))=\sum_{v\in M_{K}}n_{v}G^{+}_{v,n}(x)$ from Eqn. (6.1).

• •

We may take $c_{f,v}=0$ for any $v\not\in S$ from Proposition 4.3 and Proposition 6.2.

• •

$\sum_{v\in M_{K}}n_{v}(\log^{+}\|x\|_{v}+c_{f,v})=h(x)+\sum_{v\in S}n_{v}c_{f,v}<+\infty$.

Lebesgue’s dominant convergence theorem then implies that $\sum_{v\in M_{K}}n_{v}G^{+}_{v,n}(x)$ converges as $n\to+\infty$ and that

[TABLE]

This completes the proof of (1)(2).

(3)(4)(5) First we have

[TABLE]

On the other hand, we have

• •

$\min\{c^{+}_{\varepsilon,\delta,v},c^{-}_{\varepsilon,\delta,v}\}+\log^{+}\|x\|\leq G_{f,v}(x)+G_{f^{-1},v}(x)$ from Lemma 2.8, Eqn. (2.12) and Theorem 5.1.

• •

For any $v\not\in S$, we may take $\varepsilon=1,\delta=1$ and $\min\{c^{+}_{1,1,v},c^{-}_{1,1,v}\}=0$ from Proposition 4.3 and Proposition 6.2.

Then

[TABLE]

Eqns. (6.3) and (6.4) give (3)(ii). For the rest of the proof, see [7, Theorem 4.2(2)(3)(4)].   $\Box$

Remark 6.4**.**

Theorem 6.3(1) shows that $\{\frac{1}{d^{n}}h(f^{n}(x))\}_{n=0}^{+\infty}$ and $\{\frac{1}{d_{-}^{n}}h(f^{-n}(x))\}_{n=0}^{+\infty}$ are convergent sequences, which gives an improvement of [7], since we replace $\limsup$ by $\lim$ in the definition of $\widehat{h}^{\pm}_{f}$.

We now introduce another function

[TABLE]

for $x\in{\mathbb{A}}^{N}(\overline{K})$. The next proposition shows that $\widetilde{h}_{f}$ also behaves well relative to $f$.

Proposition 6.5**.**

1. (1)

$\widetilde{h}_{f}=h+O(1)$* on ${\mathbb{A}}^{N}(\overline{K})$.* 2. (2)

For $x\in{\mathbb{A}}^{N}(\overline{K})$, we have $\widetilde{h}_{f}(x)=0$ if and only if $\widehat{h}_{f}(x)=0$.

Proof. (1) We use the notation of the proof of Theorem 6.3. By Lemma 1.3, Eqn. (5.4), Lemma 2.8, Eqn. (2.12) and Theorem 5.1, we have

[TABLE]

Summing up over all places $v$, we get

[TABLE]

Since we have $c_{f,v}=c_{f^{-1},v}=c^{+}_{\varepsilon,\delta,v}=c^{-}_{\varepsilon,\delta,v}=0$ except for finitely many $v$ (indeed for every $v\not\in S$), this gives the assertion.

(2) Since $G_{f,v}$ and $G_{f^{-1},v}$ are non-negative functions, we see that $\widetilde{h}_{f}(x)=0$ if and only if $G_{f,v}(x)=G_{f^{-1},v}(x)=0$ if and only if $\widehat{h}_{f}(x)=0$.   $\Box$

7. Arithmetic properties of regular polynomial automorphisms

In this section, we give some applications of local and global canonical height functions. The first application is the following theorem on the usual height function (see [7, §4], [18, Conjecture 3], [19, Conjecture 7.18]), which is independently obtained by Lee [11] via a different method (via a global method based on the effectiveness of a certain divisor as in the case of $N=2$ in [7]).

Theorem 7.1**.**

Let $f:{\mathbb{A}}^{N}\to{\mathbb{A}}^{N}$ be a regular polynomial automorphism over a number field $K$. Let $d$ and $d_{-}$ be the degrees of $f$ and $f^{-1}$. Then we have the following.

1. (1)

There exists a constant $c\geq 0$ such that

[TABLE]

for all $x\in{\mathbb{A}}^{N}(\overline{K})$. 2. (2)

One has

[TABLE]

Proof. (1) We set

[TABLE]

Claim 7.1.1**.**

For all $x\in{\mathbb{A}}^{N}(\overline{K})$, we have

[TABLE]

We first show that Claim 7.1.1 implies (1). Indeed, we assume Claim 7.1.1. Then summing up over all $v$, we have

[TABLE]

Since $\widetilde{h}_{f}=h+O(1)$ by Proposition 6.5(1), Eqn. (7.3) yields (1).

Recall that $G_{f,v}$ and $G_{f^{-1},v}$ are non-negative. We will show (7.2) according to the four cases: $\frac{1}{dd_{-}}G_{f,v}(x)\gtrless G_{f^{-1},v}(x)$ and $\frac{1}{dd_{-}}G_{f^{-1},v}(x)\gtrless G_{f,v}(x)$. The first case treats $(\geq,\leq)$, and then $(\leq,\leq)$, $(\geq,\geq)$ and $(\leq,\geq)$ in this order.

Case 1: Suppose that $\frac{1}{dd_{-}}G_{f,v}(x)\geq G_{f^{-1},v}(x)$. It follows that $G_{f,v}(x)\geq dd_{-}G_{f^{-1},v}(x)\geq\frac{1}{dd_{-}}G_{f^{-1},v}(x)$. In this case, we have

[TABLE]

and similarly

[TABLE]

Further, the assumption $\frac{1}{dd_{-}}G_{f,v}(x)\geq G_{f^{-1},v}(x)$ implies that $G_{f,v}(x)\geq G_{f^{-1},v}(x)$. Thus $\widetilde{G}_{f,v}(x)=G_{f,v}(x)$. Putting these together, we get

[TABLE]

Case 2: Suppose that $G_{f,v}(x)\geq\frac{1}{dd_{-}}G_{f^{-1},v}(x)$ and $\frac{1}{dd_{-}}G_{f,v}(x)\leq G_{f^{-1},v}(x)$. In this case, we have $\widetilde{G}_{f,v}(f(x))=dG_{f,v}(x)$ and $\widetilde{G}_{f,v}(f^{-1}(x))=d_{-}G_{f^{-1},v}(x)$.

Subcase 2-1: Suppose that $G_{f,v}(x)\geq G_{f^{-1},v}(x)$. Then, using $\frac{1}{dd_{-}}G_{f,v}(x)\leq G_{f^{-1},v}(x)$, we have

[TABLE]

Subcase 2-2: Suppose that $G_{f^{-1},v}(x)\geq G_{f,v}(x)$. Then, using $G_{f,v}(x)\geq\frac{1}{dd_{-}}G_{f^{-1},v}(x)$,

[TABLE]

Case 3: Suppose that $G_{f,v}(x)\leq\frac{1}{dd_{-}}G_{f^{-1},v}(x)$ and $\frac{1}{dd_{-}}G_{f,v}(x)\geq G_{f^{-1},v}(x)$. Then $G_{f^{-1},v}(x)\leq\frac{1}{dd_{-}}G_{f,v}(x)\leq\left(\frac{1}{dd_{-}}\right)^{2}G_{f^{-1},v}(x)$. It follows that $G_{f,v}(x)=G_{f^{-1},v}(x)=0$. Then

[TABLE]

and similarly we have $\widetilde{G}_{f,v}(f^{-1}(x))=0$ and $\widetilde{G}_{f,v}(x)=0$. We get

[TABLE]

Case 4: Suppose that $G_{f,v}(x)\leq\frac{1}{dd_{-}}G_{f^{-1},v}(x)$. It follows that $G_{f^{-1},v}(x)\geq dd_{-}G_{f,v}(x)\geq\frac{1}{dd_{-}}G_{f,v}(x)$. We can show (7.2) as in Case 1, exchanging the roles of $G_{f,v}$ and $G_{f^{-1},v}$.

This completes the proof of Claim 7.1.1, hence the proof of Theorem 7.1(1).

(2) From (1), we obtain

[TABLE]

On the other hand, it is shown in [7, Proposition 4.4] that, for any polynomial automorphism $f:{\mathbb{A}}^{N}\to{\mathbb{A}}^{N}$, one has

[TABLE]

Combining these two inequalities gives the assertion.   $\Box$

Remark 7.2**.**

It is shown in [7, Theorem 4.4] that the equality (7.1) holds in dimension $N=2$ for regular polynomial automorphisms. Theorem 7.1(2) asserts that the equality holds in any dimension $N\geq 2$ for regular polynomial automorphisms.

Theorem 6.3 recovers the following theorem of Marcello [12] on $f$-periodic points.

Corollary 7.3** ([12]).**

Let $f:{\mathbb{A}}^{N}\to{\mathbb{A}}^{N}$ be a regular polynomial automorphism over a number field $K$. Then the set of $f$-periodic points in ${\mathbb{A}}^{N}(\overline{K})$ is a set of bounded height. In particular, for any integer $D\geq 1$, the set

[TABLE]

is finite.

Proof. By Theorem 6.3(3-ii) and (5), $\widehat{h}_{f}$ satisfies $\widehat{h}_{f}\gg\ll h$, and a point $x\in{\mathbb{A}}^{N}(\overline{K})$ is $f$-periodic if and only if $\widehat{h}_{f}(x)=0$. Thus we get the assertion.   $\Box$

For a non $f$-periodic point $x$, let $O_{f}(x):=\{f^{n}(x)\mid n\in{\mathbb{Z}}\}$ denote the $f$-orbit of $x$. We define the canonical height of the orbit $O_{f}(x)$ by

[TABLE]

whose value depends only on the orbit $O_{f}(x)$ and not the particular choice of the point $x$ in the orbit by Theorem 6.3. The next corollary gives a refinement of [13, Corollaire B].

Corollary 7.4**.**

Let $f:{\mathbb{A}}^{N}\to{\mathbb{A}}^{N}$ be a regular polynomial automorphism over a number field $K$. Let $d$ and $d_{-}$ be the degrees of $f$ and $f^{-1}$. Then for any infinite orbit $O_{f}(x)$,

[TABLE]

as $T\to+\infty$. Here the $O(1)$ bound depends only $f$, independent of the orbit $O_{f}(x)$.

Proof. Since $f$ satisfies (7.3), we apply [7, Theorem 5.2].   $\Box$

In the rest of this section, we consider some global-to-local arithmetic properties. Suppose that $f$ is a regular polynomial automorphism. By Theorem 6.3(2)(5), $x\in{\mathbb{A}}^{N}(\overline{K})$ is $f$-periodic if and only if $G_{f,v}(x)=0$ for all $v\in M_{K}$. By Theorem 3.1 for non-Archimedean $v$ and [16, §2] for Archimedean $v$, $G_{f,v}(x)=0$ is equivalent to $\{f^{n}(x)\}_{n=0}^{+\infty}$ being bounded with respect to $\|\cdot\|_{v}$. Thus we see that $x\in{\mathbb{A}}^{N}(\overline{K})$ is $f$-periodic if and only if $\{f^{n}(x)\}_{n=0}^{+\infty}$ is bounded with respect to $\|\cdot\|_{v}$ for all $v\in M_{K}$.

This actually holds for any polynomial map $f$ (cf. [3, Corollary 6.3] for $N=1$).

Proposition 7.5**.**

Let $f:{\mathbb{A}}^{N}\to{\mathbb{A}}^{N}$ be a polynomial map over a number field $K$. For $x\in{\mathbb{A}}^{N}(\overline{K})$, the following is equivalent.

1. (i)

$x$* is $f$-periodic.* 2. (ii)

For every $v\in M_{K}$, $\{f^{n}(x)\}_{n=0}^{+\infty}$ is bounded with respect to the $v$-adic topology.

Proof. Taking a finite extension field of $K$ over which $x$ is defined if necessary, we may assume that $x$ is defined over $K$. It is obvious that (i) implies (ii). We assume (ii) and show (i). We take a finite subset $S$ of $M_{K}$ containing the set of all Archimedean absolute values such that $x$ and $f$ is defined over $(O_{K})_{S}$. Then for any $v\not\in S$, we have

[TABLE]

Since we assume (ii), there is a constant $C_{v}$ for each $v\in S$ such that

[TABLE]

Then we have

[TABLE]

Then

[TABLE]

Since the latter set is finite, the set $\{f^{n}(x)\}_{n\geq 0}$ is finite. Hence $x$ is $f$-periodic.   $\Box$