On permutations induced by tame automorphisms over finite fields

TL;DR

This paper provides formulas to determine the sign of permutations induced by tame automorphisms over finite fields, enabling easier analysis of their permutation properties based on their decompositions.

Contribution

It introduces explicit formulas for the permutation sign induced by elementary and affine automorphisms over finite fields, and combines these to analyze tame automorphisms.

Findings

Formulas for signs of elementary automorphisms

Formulas for signs of affine automorphisms

Method to determine signs of tame automorphisms

Abstract

The present paper deals with permutations induced by tame automorphisms over finite fields. The first main result is a formula for determining the sign of the permutation induced by a given elementary automorphism over a finite field. The second main result is a formula for determining the sign of the permutation induced by a given affine automorphism over a finite field. We also give a combining method of the above two formulae to determine the sign of the permutation induced by a given triangular automorphism over a finite field. As a result, for a given tame automorphism over a finite field, if we know a decomposition of the tame automorphism into a finite number of affine automorphisms and elementary automorphisms, then one can easily determine the sign of the permutation induced by the tame automorphism.

Equations

$$F(a_1,\dots ,a_n)=\left(f_1(a_1,\dots ,a_n),\dots ,f_n(a_1,\dots ,a_n)\right)$$

$$\begin{array}[]{ccccc}\pi:&\mathop{\mathrm{MA}_{n}}(k)&\rightarrow&\mathop{\mathrm{Maps}}({k^{n}},{k^{n}}).\end{array}$$

$$\begin{array}{rl}{E}_{a_i}& =(X_1,\dots ,X_{i-1},X_i+a_i,X_{i+1},\dots ,X_n),\\ & a_i\in k[X_1,\dots ,\widehat{X_i},\dots ,X_n]=k[X_1,\dots ,X_{i-1},X_{i+1},\dots ,X_n],\end{array}$$

$$\begin{array}{rl}{J}_{a,f}& =\left(a_1{X}_1+f_1(X_2,\dots ,X_n),a_2{X}_2+f_2(X_3,\dots ,X_n),\dots ,a_n{X}_n+f_n\right),\\ & a_i\in k\left(i=1,\dots ,n\right),f_i\in k[X_{i+1},\dots ,X_n]\left(i=1,\dots ,n-1\right),f_n\in k,\end{array}$$

$$\begin{array}{rl}{\mathrm{BA}}_n(k)& :=\{J_{a,f}\in {\mathrm{GA}}_n(k)\mid a_i\in k\left(i=1,\dots ,n\right),\\ & f_i\in k[X_{i+1},\dots ,X_n]\left(i=1,\dots ,n-1\right),f_n\in k\}.\end{array}$$

$$\begin{array}[]{ccccc}{\pi}_{q}:&\mathop{\mathrm{MA}_{n}}(\mathbb{F}_{q})&\rightarrow&\mathop{\mathrm{Maps}}({\mathbb{F}_{q}^{n}},{\mathbb{F}_{q}^{n}}).\end{array}$$

$$\begin{array}[]{ccccc}{\pi}_{q}:&\mathop{\mathrm{GA}_{n}}(\mathbb{F}_{q})&\rightarrow&\mathop{\mathrm{Sym}}({\mathbb{F}_{q}^{n}}),\end{array}$$

$$\mathrm{sgn}:\mathrm{Sym}(S)\to \{\pm 1\}$$

$${\mathrm{Aff}}_n\left(k\right)\cong G{L}_n\left(k\right)⋉k^n.$$

$$\sigma$$

$$\tau \circ \sigma$$

$$\begin{split}\begin{array}[]{cccl}S&\longrightarrow&S&\\ \rotatebox{90.0}{$\in$}&&\rotatebox{90.0}{$\in$}&\\ s_{i}&\longmapsto&s_{i}&\text{if }i\not\in\{{i_{1}},\ldots,{i_{r}}\}\\ s_{i_{1}}&\longmapsto&s_{i_{2}}&\\ &\vdots&&\\ s_{i_{r-1}}&\longmapsto&s_{i_{r}}&\\ s_{i_{r}}&\longmapsto&s_{i_{1}}&\\ \end{array}\end{split}$$

$$E_{a_i}^{(q)}=(X_1,\dots ,X_{i-1},X_i+a_i,X_{i+1},\dots ,X_n)\in {\mathrm{EA}}_n\left({\mathbb{F}}_q\right),$$

$$\mathrm{sgn}\left({\pi }_q\left(E_{a_i}^{(q)}\right)\right)=1.$$

$$\begin{split}E_{c_{i},\mathbf{e}}^{(q)}&:=\Biggl{(}{X_{1}},\ldots,{X_{i-1}},{X_{i}}+{c_{i}}\prod_{\begin{subarray}{c}1\leq j\leq n\\ j\neq i\end{subarray}}{X_{j}^{e_{j}}},{X_{i+1}},\ldots,{X_{n}}\Biggr{)}\\ &=\Biggl{(}{X_{1}},\ldots,{X_{i-1}},{X_{i}}+{c_{i}}\prod_{j\in N_{\hat{i}}}{X_{j}^{e_{j}}},{X_{i+1}},\ldots,{X_{n}}\Biggr{)}\in{\mathop{\mathrm{EA}}}_{n}\left(\mathbb{F}_{q}\right).\end{split}$$

$$\begin{split}\begin{array}[]{ccccl}\lambda_{{c_{i}},\mathbf{e},\mathbf{y}}^{(q)}:&\mathbb{F}_{q}^{n}&\longrightarrow&\mathbb{F}_{q}^{n}&\\ &\rotatebox{90.0}{$\in$}&&\rotatebox{90.0}{$\in$}&\\ &({x_{1}},\ldots,{x_{n}})&\longmapsto&({x_{1}},\ldots,{x_{n}})&\text{if }{\exists}j\in N_{\hat{i}}\text{ s.t. }{x_{j}}\neq{y_{j}},\\ &({x_{1}},\ldots,{x_{n}})&\longmapsto&{E_{c_{i},\mathbf{e}}^{(q)}}({x_{1}},\ldots,{x_{n}})&{x_{j}}={y_{j}}\text{ for all }j\in N_{\hat{i}}.\end{array}\end{split}$$

$${\lambda }_{c_i,\mathbf{e},\mathbf{y}}^{(q)}(x_1,\dots ,x_n)=(x_1,\dots ,x_n)$$

$${\lambda }_{c_i,\mathbf{e},\mathbf{y}}^{(q)}(x_1,\dots ,x_n)=E_{c_i,\mathbf{e}}^{(q)}(x_1,\dots ,x_n)$$

$$B\left({\lambda }_{c_i,\mathbf{e},\mathbf{y}}^{(q)}\right):=\left\{(x_1,\dots ,x_n)\in {\mathbb{F}}_q^n|{\lambda }_{c_i,\mathbf{e},\mathbf{y}}^{(q)}(x_1,\dots ,x_n)\ne (x_1,\dots ,x_n)\right\}.$$

$$B\left({\lambda }_{c_i,\mathbf{e},\mathbf{y}}^{(q)}\right)\cap B\left({\lambda }_{c_i,\mathbf{e},{\mathbf{y}}^{\prime }}^{(q)}\right)=\varnothing$$

$$\begin{array}{rl}{\pi }_q\left(E_{c_i,\mathbf{e}}^{(q)}\right)& ={\pi }_q\left(\prod _{y_1,\dots ,y_{i-1},y_{i+1},\dots ,y_n\in {\mathbb{F}}_q}{\lambda }_{c_i,\mathbf{e},\mathbf{y}}^{(q)}\right)\\ & =\prod _{y_1,\dots ,y_{i-1},y_{i+1},\dots ,y_n\in {\mathbb{F}}_q}{\pi }_q\left({\lambda }_{c_i,\mathbf{e},\mathbf{y}}^{(q)}\right),\end{array}$$

$$M_1=\chi {\left(c_i\right)}^{\ell }\times {\prod }_{j\in N_{\hat{i}}}\left(q-\delta \left(e_j\right)\right).$$

$$C_y:=\{y^{\prime }\in {\mathbb{F}}_q\mid y\sim y^{\prime }\}.$$

$$♯\mathcal{R}=q/p=p^m/p=p^{m-1}.$$

$$\mathcal{R}:=\{w_1,\dots ,w_{p^{m-1}}\}\subset {\mathbb{F}}_q.$$

$$\begin{split}\begin{array}[]{ccccl}\lambda_{{c_{i}},\mathbf{e},{\mathbf{y}_{w_{s}}}}^{(q)}:&\mathbb{F}_{q}^{n}&\longrightarrow&\mathbb{F}_{q}^{n}&\\ &\rotatebox{90.0}{$\in$}&&\rotatebox{90.0}{$\in$}&\\ &({x_{1}},\ldots,{x_{n}})&\longmapsto&({x_{1}},\ldots,{x_{n}})&\\ &&&&\text{if }{\exists}j\in N_{\hat{i}}\text{ s.t. }{x_{j}}\neq{y_{j}},\text{ or }{x_{i}}\not\in C_{w_{s}},\\ &({x_{1}},\ldots,{x_{n}})&\longmapsto&{E_{c_{i},\mathbf{e}}^{(q)}}({x_{1}},\ldots,{x_{n}})&\\ &&&&{x_{j}}={y_{j}}\text{ for all }j\in N_{\hat{i}}\text{ and }{x_{i}}\in C_{w_{s}}.\\ \end{array}\end{split}$$

$$\mathrm{sgn}\left({\pi }_q\left({\lambda }_{c_i,\mathbf{e},{\mathbf{y}}_{w_s}}^{(q)}\right)\right)={\left(-1\right)}^{p-1}.$$

$${\pi }_q\left({\lambda }_{c_i,\mathbf{e},\mathbf{y}}^{(q)}\right)={\pi }_q\left(\prod _{s=1}^{p^{m-1}}{\lambda }_{c_i,\mathbf{e},{\mathbf{y}}_{w_s}}^{(q)}\right)=\prod _{s=1}^{p^{m-1}}{\pi }_q\left({\lambda }_{c_i,\mathbf{e},{\mathbf{y}}_{w_s}}^{(q)}\right).$$

$$M_2=\chi {\left(c_i\right)}^{\ell }\times p^{m-1}.$$

$$\mathrm{sgn}\left({\pi }_q\left(E_{c_i,\mathbf{e}}^{(q)}\right)\right)={\left(-1\right)}^M,$$

Full text

1. Introduction

Let $k$ be a field and let $k[{X_{1}},\ldots,{X_{n}}]$ be a polynomial ring in $n$ indeterminates over $k$. An $n$-tuple of polynomials $F=({f_{1}},\ldots,{f_{n}})$ is called a polynomial map, when ${f_{i}}$ belongs to $k[{X_{1}},\ldots,{X_{n}}]$ for all $i$, $1\leq i\leq n$. A polynomial map $F=({f_{1}},\ldots,{f_{n}})$ can be viewed as a map from ${k^{n}}$ to itself by defining

[TABLE]

for $({a_{1}},\ldots,{a_{n}})\in{k^{n}}$. The set of polynomial maps over $k$ and the set of maps from ${k^{n}}$ to itself are denoted by $\mathop{\mathrm{MA}_{n}}(k)$ and $\mathop{\mathrm{Maps}}({k^{n}},{k^{n}})$, respectively. Then by (1.1), we see that there exists a natural map

[TABLE]

$\mathop{\mathrm{MA}_{n}}(k)$ (resp. $\mathop{\mathrm{Maps}}({k^{n}},{k^{n}})$) is a semi-group with respect to the composition of polynomial maps (resp. maps). Moreover, these two semi-groups are monoids with the identity maps as neutral elements, and the map $\pi$ is a monoid homomorphism.

Let $\mathop{\mathrm{GA}_{n}}(k)$ be the subset of invertible elements in $\mathop{\mathrm{MA}_{n}}(k)$. An element in $\mathop{\mathrm{GA}_{n}}(k)$ is called a polynomial automorphism. A polynomial automorphism $F=({f_{1}},\ldots,{f_{n}})$ is said to be affine when $\deg{f_{i}}=1$ for $i=1,\ldots,n$. A polynomial map $E_{a_{i}}$ of the form

[TABLE]

is also a polynomial automorphism since ${E_{a_{i}}^{-1}}=E_{-a_{i}}$. A polynomial automorphism $E_{a_{i}}$ of form (1.2) is said to be elementary. Let $\mathop{\mathrm{Aff}}_{n}(k)$ denote the set of all affine automorphisms, and let $\mathop{\mathrm{EA}}_{n}(k)$ denote the set of all elementary automorphisms. $\mathop{\mathrm{Aff}}_{n}(k)$ and $\mathop{\mathrm{EA}}_{n}(k)$ are subgroups of $\mathop{\mathrm{GA}_{n}}(k)$. We put $\mathop{\mathrm{TA}_{n}}(k):=\langle\mathop{\mathrm{Aff}}_{n}(k),\mathop{\mathrm{EA}}_{n}(k)\rangle$, where $\langle{H_{1}},{H_{2}}\rangle$ is a subgroup of a group $G$ generated by two subgroups ${H_{1}},{H_{2}}\subset G$. Then $\mathop{\mathrm{TA}_{n}}(k)$ is also a subgroup of $\mathop{\mathrm{GA}_{n}}(k)$ and is called the tame subgroup. If ${\tau}\in\mathop{\mathrm{TA}_{n}}(k)$ then ${\tau}$ is called tame automorphism, and otherwise (i.e., $\tau\in\mathop{\mathrm{GA}_{n}}(k)\setminus\mathop{\mathrm{TA}_{n}}(k)$) $\tau$ is called wild automorphism. A polynomial automorphism $J_{a,f}$ of the form

[TABLE]

is called de Jonquières automorphism (or triangular automorphism). From the definition, triangular automorphism is trivially polynomial automorphism. The set of all elements of the form $J_{a,f}$ is denoted by $\mathop{\mathrm{BA}_{n}}(k)$:

[TABLE]

$\mathop{\mathrm{BA}_{n}}(k)$ is a subgroup of $\mathop{\mathrm{GA}_{n}}(k)$. $\mathop{\mathrm{BA}_{n}}(k)$ is also a subgroup of $\mathop{\mathrm{TA}_{n}}(k)$, and it is known that $\mathop{\mathrm{TA}_{n}}(k)=\langle\mathop{\mathrm{Aff}}_{n}(k),\mathop{\mathrm{BA}_{n}}(k)\rangle$ (cf. [2, Exercises for § 5.1 – 1, 2]). The Tame Generators Problem asks whether $\mathop{\mathrm{GA}_{n}}(k)=\mathop{\mathrm{TA}_{n}}(k)$, and is related to the famous Jacobian conjecture.

Tame Generators Problem**.**

$\mathop{\mathrm{GA}_{n}}(k)=\mathop{\mathrm{TA}_{n}}(k)$? **

For any field $k$, we denote the characteristic of the field $k$ by $p=\text{char}(k)$. If $k$ is a finite field $\mathbb{F}_{q}$ with $q$ elements ($p=\text{char}(\mathbb{F}_{q})$, $q=p^{m}$, and $m\geq 1$), we use the symbol ${\pi}_{q}$ instead of ${\pi}$:

[TABLE]

The map ${\pi}_{q}$ induces a group homomorphism

[TABLE]

where $\mathop{\mathrm{Sym}}(S)$ is a symmetric group on a finite set $S$. Let

[TABLE]

be the sign function. The sign function $\mathop{\mathrm{sgn}}$ is a group homomorphism, and $\mathop{\mathrm{Ker}}(\mathop{\mathrm{sgn}})=\mathop{\mathrm{Alt}}(S)$, where $\mathop{\mathrm{Alt}}(S)$ is the alternating group on $S$. Recall that for any subgroup $G\subseteq\mathop{\mathrm{GA}_{n}}(\mathbb{F}_{q})$, ${\pi}_{q}\left(G\right)$ is a subgroup of $\mathop{\mathrm{Sym}}({\mathbb{F}_{q}^{n}})$.

In the case where $k=\mathbb{F}_{q}$, we can consider a slightly different problem from the Tame Generators Problem, namely, it is natural to investigate the subgroup ${\pi}_{q}\left(G\right)$ of $\mathop{\mathrm{Sym}}({\mathbb{F}_{q}^{n}})$. This problem has first been investigated in the case $G=\mathop{\mathrm{TA}_{n}}(\mathbb{F}_{q})$ by Maubach [3]. Indeed, Maubach proved the following theorem ([3, Theorem 2.3]) and proposed a following problem ([4, page 3, Problem]):

Theorem 1**.**

([3, Theorem 2.3])*  If $n\geq 2$, then ${\pi}_{q}(\mathop{\mathrm{TA}_{n}}(\mathbb{F}_{q}))=\mathop{\mathrm{Sym}}({\mathbb{F}_{q}^{n}})$ if $q$ is odd or $q=2$. If $q={2^{m}}$ where $m\geq 2$ then ${\pi}_{q}(\mathop{\mathrm{TA}_{n}}(\mathbb{F}_{q}))=\mathop{\mathrm{Alt}}({\mathbb{F}_{q}^{n}})$.*

Question 1**.**

([4, page 3, Problem])*  *For $q=2^{m}$ and $m\geq 2$, do there exist polynomial automorphisms such that the permutations induced by the polynomial automorphisms belong to $\mathop{\mathrm{Sym}}({\mathbb{F}_{q}^{n}})\setminus\mathop{\mathrm{Alt}}({\mathbb{F}_{q}^{n}})$? **

If there exists $F\in\mathop{\mathrm{GA}_{n}}(\mathbb{F}_{2^{m}})$ such that ${\pi}_{2^{m}}(F)\in\mathop{\mathrm{Sym}}({\mathbb{F}_{2^{m}}^{n}})\setminus\mathop{\mathrm{Alt}}({\mathbb{F}_{2^{m}}^{n}})$, then we must have $F\in\mathop{\mathrm{GA}_{n}}(\mathbb{F}_{2^{m}})\setminus\mathop{\mathrm{TA}_{n}}(\mathbb{F}_{2^{m}})$, namely, $F$ is a wild automorphism. Thus, Question 1 is a quite important problem for the Tame Generators Problem in positive characteristic. Furthermore, we refer the reader to [5, Section 1.2] for several questions related to [3, Theorem 2.3]. The present paper deals with permutations induced by tame automorphisms over finite fields. We address the following questions related to [3, Theorem 2.3] which is a little different from the questions in [5, Section 1.2].

Question 2**.**

For a given tame automorphism ${\phi}\in\mathop{\mathrm{TA}_{n}}(\mathbb{F}_{q})$, how to determine the sign of the permutation induced by ${\phi}$? (how to determine $\mathop{\mathrm{sgn}}({\pi}_{q}({\phi}))$?) **

Question 3**.**

Suppose that $G$ is a subgroup of $\mathop{\mathrm{GA}_{n}}(\mathbb{F}_{q})$. What are sufficient conditions on $G$ such that the inclusion relation ${\pi}_{q}\left(G\right)\subset\mathop{\mathrm{Alt}}({\mathbb{F}_{q}^{n}})$ holds? **

Question 2 seems to be natural since if $q\neq{2^{m}}$ ($m\geq 2$), one can not determine the sign of the permutation induced by a given tame automorphism over a finite field from [3, Theorem 2.3]. The information about sign of the permutations might be useful for studying the Tame Generators Problem, Question 1, or other related questions such as [4, page 5, Conjecture 4.1], [4, page 5, Conjecture 4.2] and so on. Question 3 also seems to be natural since if $q\neq{2^{m}}$ ($m\geq 2$) and $G\neq\mathop{\mathrm{TA}_{n}}(\mathbb{F}_{q})$, one can not obtain any sufficient condition for the inclusion relation ${\pi}_{q}\left(G\right)\subset\mathop{\mathrm{Alt}}({\mathbb{F}_{q}^{n}})$ from [3, Theorem 2.3].

The contributions of the present paper are as follows. The first main result is a formula for determining the sign of the permutation induced by a given elementary automorphism over a finite field (Section 3, Main Theorem 1). Our method to determine the sign of the permutation induced by a given elementary automorphism over a finite field, is based on group theory. As a consequence of Main Theorem 1, one can derive a sufficient condition for the inclusion relation ${\pi}_{q}\left({\mathop{\mathrm{EA}}}_{n}\left(\mathbb{F}_{q}\right)\right)\subset\mathop{\mathrm{Alt}}({\mathbb{F}_{q}^{n}})$ (Section 3, Corollary 1). The second main result is a formula for determining the sign of the permutation induced by a given affine automorphism over a finite field (Section 4, Main Theorem 2). Our method to determine the sign of the permutation induced by a given affine automorphism over a finite field, is based on linear algebra. As a consequence of Main Theorem 2, one can also derive a sufficient condition for the inclusion relation ${\pi}_{q}\left({\mathop{\mathrm{Aff}}}_{n}\left(\mathbb{F}_{q}\right)\right)\subset\mathop{\mathrm{Alt}}({\mathbb{F}_{q}^{n}})$ (Section 4, Corollary 2). Section 5 gives a combining method of Main Theorem 1 and Main Theorem 2 to determine the sign of the permutation induced by a given triangular automorphism over a finite field (Section 5, Corollary 3). As a result, for a given tame automorphism over a finite field, if we know a decomposition of the tame automorphism into a finite number of affine automorphisms and elementary automorphisms, then one can easily determine the sign of the permutation induced by the tame automorphism (Section 5, Corollary 5).

The rest of this paper is organized as follows. In Section 2, we fix our notation. In Section 3, we give a method to determine the sign of the permutation induced by a given elementary automorphism over a finite field. In Section 4, we give a method to determine the sign of the permutation induced by a given affine automorphism over a finite field. Section 5 deals with how to determine the sign of the permutation induced by a given triangular automorphism and a given tame automorphism over a finite field.

2. Notation

Throughout this paper, we use the following notation. For any field $k$, we denote the characteristic of the field $k$ by $p=\text{char}(k)$. We denote the multiplicative group of a field $k$ by $k^{*}=k\setminus\{0\}$. We use the symbols $\mathbb{Z}$, $\mathbb{C}$, $\mathbb{F}_{q}$ to represent the rational integer ring, the field of complex numbers, and a finite field with $q$ elements ($p=\text{char}(\mathbb{F}_{q})$, $q=p^{m}$, $m\geq 1$). We denote the set of non-negative integers by $\mathbb{Z}_{\geq 0}$. For a group $G$ and $g\in G$, $\text{ord}_{G}(g)$ is the order of $g$, namely, $\text{ord}_{G}(g)$ is the smallest non-negative integer $x$ that holds $g^{x}=1_{G}$, where $1_{G}$ is the identity element in the group $G$. We denote by $GL_{n}\left(k\right)$ the set of invertible matrices with entries in $k$.

The polynomial ring $n$ indeterminates over $k$ is denoted by $k[{X_{1}},\ldots,{X_{n}}]$. We denote the polynomial ring $k[{X_{1}},\ldots,{X_{i-1}},{X_{i+1}},\ldots,{X_{n}}]$ (omit the indeterminate ${X_{i}}$) by $k[{X_{1}},\ldots,\hat{X_{i}},\ldots,{X_{n}}]$. Let $\mathop{\mathrm{MA}_{n}}(k)$ be the set of polynomial maps over $k$. $\mathop{\mathrm{MA}_{n}}(k)$ is a monoid with respect to the composition of polynomial maps, and the neutral elements of the monoid $\mathop{\mathrm{MA}_{n}}(k)$ is the identity map. We denote the subset of invertible elements in $\mathop{\mathrm{MA}_{n}}(k)$, the set of all affine automorphisms, the set of all elementary automorphisms, and the set of all triangular automorphisms by $\mathop{\mathrm{GA}_{n}}(k)$, $\mathop{\mathrm{Aff}}_{n}(k)$, $\mathop{\mathrm{EA}}_{n}(k)$, and $\mathop{\mathrm{BA}_{n}}(k)$, respectively. $\mathop{\mathrm{GA}_{n}}(k)$ is a group with respect to the composition of polynomial automorphisms, and $\mathop{\mathrm{Aff}}_{n}(k)$, $\mathop{\mathrm{EA}}_{n}(k)$ are subgroups of $\mathop{\mathrm{GA}_{n}}(k)$. Recall that

[TABLE]

For a finite set $S$, we denote the cardinality of $S$ by ${\sharp}S$, and denote the symmetric group on $S$ and the alternating group on $S$ by $\mathop{\mathrm{Sym}}(S)$ and $\mathop{\mathrm{Alt}}(S)$, respectively. Let $S=\{{s_{1}},\ldots,{s_{n}}\}$, $\sharp S=n$, and $\sigma\in\mathop{\mathrm{Sym}}(S)$. For $\sigma\in\mathop{\mathrm{Sym}}(S)$,

[TABLE]

means that $\sigma\left({s_{i}}\right)={s_{\sigma\left(i\right)}}$ for $1\leq i\leq n$. For $\sigma,\tau\in\mathop{\mathrm{Sym}}(S)$, we denote by $\tau\circ\sigma$ the composition

[TABLE]

The permutation on $S$ defined by

[TABLE]

is called the cycle of length $r$ and is denoted by $\left({i_{1}}\;{i_{2}}\;\ldots\;{i_{r}}\right)$. Let ${\delta}:\mathbb{R}\rightarrow\mathbb{R}$ be a function satisfying that ${\delta}\left(x\right)=0$ when $x=0$, and ${\delta}\left(x\right)=1$ when $x\neq 0$. Let ${\chi}:\mathbb{F}_{q}^{*}\rightarrow\mathbb{C}^{*}$ be a non-trivial multiplicative character of order $\ell$. We extend ${\chi}$ to $\mathbb{F}_{q}$ by defining ${\chi}\left(0\right)=0$.

3. Sign of permutations induced by elementary automorphisms

In this section, we consider the sign of permutations induced by elementary automorphisms over finite fields. The main result of this section is as follows.

Main Theorem 1**.**

(Sign of elementary automorphisms)  Suppose that $E_{a_{i}}^{(q)}$ is an elementary automorphism over a finite field, namely,

[TABLE]

and ${a_{i}}\in\mathbb{F}_{q}[{X_{1}},\ldots,\hat{X_{i}},\ldots,{X_{n}}]$. If $q$ is odd or $q=2^{m}$, $m\geq 2$ then we have ${\pi}_{q}\left(E_{a_{i}}^{(q)}\right)\in\mathop{\mathrm{Alt}}({\mathbb{F}_{q}^{n}})$. Namely, if $q$ is odd or $q=2^{m}$, $m\geq 2$ then

[TABLE]

If $q=2$ then $\mathop{\mathrm{sgn}}\left({\pi}_{q}\left(E_{a_{i}}^{(q)}\right)\right)$ depends only on the number of monomials of the form $cX_{1}^{e_{1}}{\cdots}X_{i-1}^{e_{i-1}}X_{i+1}^{e_{i+1}}{\cdots}X_{n}^{e_{n}}$ with $c\in\mathbb{F}_{q}^{*}$ and ${e_{1}},\ldots,{e_{n}}\geq 1$ appearing in the polynomial ${a_{i}}$. More precisely, if $q=2$ then $\mathop{\mathrm{sgn}}\left({\pi}_{q}\left(E_{a_{i}}^{(q)}\right)\right)=\left(-1\right)^{M_{a_{i}}}$, where $M_{a_{i}}$ is the number of monomials of the form $cX_{1}^{e_{1}}{\cdots}X_{i-1}^{e_{i-1}}X_{i+1}^{e_{i+1}}{\cdots}X_{n}^{e_{n}}$ with $c\in\mathbb{F}_{q}^{*}$ and ${e_{1}},\ldots,{e_{n}}\geq 1$ appearing in the polynomial ${a_{i}}$.

Proof.

It suffices to assume that ${a_{i}}\in\mathbb{F}_{q}[{X_{1}},\ldots,\hat{X_{i}},\ldots,{X_{n}}]$ is a monomial (Similar discussion can be found in [3, page 96, Proof of Theorem 5.2.1]). Let ${c_{i}}\in\mathbb{F}_{q}$, $N_{\hat{i}}:=\{1,\ldots,i-1,i+1,\ldots,n\}$, and $\mathbf{e}:=\left({e_{1}},\ldots,\hat{e_{i}},\ldots,{e_{n}}\right)\in\mathbb{Z}_{\geq 0}^{n-1}$ (${e_{1}},\ldots,\hat{e_{i}},\ldots,{e_{n}}\geq 0$). We put ${a_{i}}={c_{i}}\prod_{\begin{subarray}{c}1\leq j\leq n\\ j\neq i\end{subarray}}{X_{j}^{e_{j}}}$ and

[TABLE]

In the following, we determine the value of $\mathop{\mathrm{sgn}}\left({\pi}_{q}\left(E_{c_{i},\mathbf{e}}^{(q)}\right)\right)$ by decomposing the permutation ${\pi}_{q}\left(E_{c_{i},\mathbf{e}}^{(q)}\right)$ as a product of transpositions. Let ${y_{1}},\ldots,{y_{i-1}},{y_{i+1}},\ldots,{y_{n}}$ be elements of $\mathbb{F}_{q}$. We put $\mathbf{y}=\left({y_{1}},\ldots,{y_{i-1}},{y_{i+1}},\ldots,{y_{n}}\right)\in\mathbb{F}_{q}^{n-1}$. For $\mathbf{y}\in\mathbb{F}_{q}^{n-1}$, we define the map $\lambda_{{c_{i}},\mathbf{e},\mathbf{y}}^{(q)}$ as follows:

[TABLE]

It follows from the definition of the map $\lambda_{{c_{i}},\mathbf{e},\mathbf{y}}^{(q)}$ that

[TABLE]

for each $({x_{1}},\ldots,{x_{n}})\in\mathbb{F}_{q}^{n}\setminus\{\left({y_{1}},\ldots,{y_{i-1}},y,{y_{i+1}},\ldots,{y_{n}}\right)\mid y\in\mathbb{F}_{q}\}$ and

[TABLE]

for each $({x_{1}},\ldots,{x_{n}})\in\{\left({y_{1}},\ldots,{y_{i-1}},y,{y_{i+1}},\ldots,{y_{n}}\right)\mid y\in\mathbb{F}_{q}\}$. Thus the map $\lambda_{{c_{i}},\mathbf{e},\mathbf{y}}^{(q)}$ is bijective and the inverse of $\lambda_{{c_{i}},\mathbf{e},\mathbf{y}}^{(q)}$ is $\lambda_{-{c_{i}},\mathbf{e},\mathbf{y}}^{(q)}$. Remark that $\lambda_{{c_{i}},\mathbf{e},\mathbf{y}}^{(q)}$ is the identity map if and only if $\mathbf{e}=\left(0,\ldots,0\right)$ and ${c_{i}}=0$, or there exists $j\in N_{\hat{i}}$ such that ${e_{j}}\neq 0$ and ${y_{j}}=0$. Put

[TABLE]

Since

[TABLE]

for any $\mathbf{y}^{\prime}=\left({y_{1}^{\prime}},\ldots,{y_{i-1}^{\prime}},{y_{i+1}^{\prime}},\ldots,{y_{n}^{\prime}}\right)\in\mathbb{F}_{q}^{n-1}$ satisfying $\mathbf{y}^{\prime}\neq\mathbf{y}$, the permutation ${\pi}_{q}\left(E_{c_{i},\mathbf{e}}^{(q)}\right)$ can be written as

[TABLE]

which is a composition of disjoint permutations on $\mathbb{F}_{q}^{n}$. We denote the number of distinct permutations other than the identity map in the right-hand side of Equation (3.3) by ${M_{1}}$. It is straightforward to check that ${M_{1}}$ satisfies the equation

[TABLE]

Next, we decompose each permutation ${\pi}_{q}\left(\lambda_{{c_{i}},\mathbf{e},\mathbf{y}}^{(q)}\right)$ as a composition of disjoint cycles on $\mathbb{F}_{q}^{n}$. In order to find such a decomposition, we define an equivalence relation ${\sim}$ on $\mathbb{F}_{q}$: $y\in\mathbb{F}_{q}$ and $y^{\prime}\in\mathbb{F}_{q}$ are equivalent if and only if there exists $l\in\{0,1,\ldots,p-1\}$ such that $y^{\prime}=y+l\left({c_{i}}\prod_{j\in N_{\hat{i}}}{y_{j}^{e_{j}}}\right)$. Note that the equivalence relation ${\sim}$ depends on the choice of ${y_{1}},\ldots,{y_{i-1}},{y_{i+1}},\ldots,{y_{n}}$. Put

[TABLE]

We now choose a complete system of representatives $\mathcal{R}$ for the above equivalence relation ${\sim}$. Since $\mathcal{R}$ is a complete system of representatives, it follows that

[TABLE]

We write

[TABLE]

For any ${w_{s}}\in\mathcal{R}$ ($1\leq s\leq{p^{m-1}}$), we set ${\mathbf{y}_{w_{s}}}:=\left({y_{1}},\ldots,{y_{i-1}},{w_{s}},{y_{i+1}},\ldots,{y_{n}}\right)$. We define the bijective map $\lambda_{{c_{i}},\mathbf{e},{\mathbf{y}_{w_{s}}}}^{(q)}$ as follows:

[TABLE]

Note that the map $\lambda_{{c_{i}},\mathbf{e},{\mathbf{y}_{w_{s}}}}^{(q)}$ is a cycle of length $p$, and the standard result from elementary group theory yields that its sign is $\left(-1\right)^{p-1}$, namely,

[TABLE]

We also remark that the map $\lambda_{{c_{i}},\mathbf{e},{\mathbf{y}_{w_{s}}}}^{(q)}$ is the identity map if and only if $\mathbf{e}=\left(0,\ldots,0\right)$ and ${c_{i}}=0$, or there exists $j\in N_{\hat{i}}$ such that ${e_{j}}\neq 0$ and ${y_{j}}=0$. If $1\leq{s_{1}},{s_{2}}\leq{p^{m-1}}$ and ${s_{1}}\neq{s_{2}}$, then we have $C_{w_{s_{1}}}\cap C_{w_{s_{2}}}=\emptyset$. This yields a decomposition of the permutation ${\pi}_{q}\left(\lambda_{{c_{i}},\mathbf{y}}\right)$ into a composition of disjoint cycles on $\mathbb{F}_{q}^{n}$:

[TABLE]

We denote the number of disjoint cycles other than the identify map in Equation (3.6) by ${M_{2}}$. By counting the number of disjoint cycles appearing in Equation (3.6), we have

[TABLE]

Now we determine the sign of the permutation ${\pi}_{q}\left(E_{c_{i},\mathbf{e}}^{(q)}\right)$. By Equation (3.3) through Equation (3.7), we obtain

[TABLE]

where

[TABLE]

If $q$ is odd (and thus $p$ is odd) or $q=2^{m}$, $m\geq 2$, we have $M\equiv 0\bmod 2$ and if $q=2$ (namely, $p=2$ and $m=1$), we have $M\equiv{\chi}\left({c_{i}}\right)^{\ell}\times{\prod}_{j\in N_{\hat{i}}}{\delta}\left({e_{j}}\right)\bmod 2$. Therefore if $q$ is odd or $q=2^{m}$, $m\geq 2$ then ${\pi}_{q}\left(E_{c_{i},\mathbf{e}}^{(q)}\right)\in\mathop{\mathrm{Alt}}({\mathbb{F}_{q}^{n}})$. Hence, we have ${\pi}_{q}\left(E_{a_{i}}^{(q)}\right)\in\mathop{\mathrm{Alt}}({\mathbb{F}_{q}^{n}})$, namely, $\mathop{\mathrm{sgn}}\left({\pi}_{q}\left(E_{a_{i}}^{(q)}\right)\right)=1$. On the other hand, if $q=2$ then $\mathop{\mathrm{sgn}}\left({\pi}_{q}\left(E_{c_{i},\mathbf{e}}^{(q)}\right)\right)=\left(-1\right)^{{\chi}\left({c_{i}}\right)^{\ell}\times{\prod}_{j\in N_{\hat{i}}}{\delta}\left({e_{j}}\right)}$. Thus, $\mathop{\mathrm{sgn}}\left({\pi}_{q}\left(E_{c_{i},\mathbf{e}}^{(q)}\right)\right)=-1$ if and only if ${c_{i}}\neq 0$ and ${\prod}_{j\in N_{\hat{i}}}{\delta}\left({e_{j}}\right)=1$, or more generally, $\mathop{\mathrm{sgn}}\left({\pi}_{q}\left(E_{a_{i}}^{(q)}\right)\right)=-1$ if and only if the number of monomials of the form $cX_{1}^{e_{1}}{\cdots}X_{i-1}^{e_{i-1}}X_{i+1}^{e_{i+1}}{\cdots}X_{n}^{e_{n}}$ with $c\in\mathbb{F}_{q}^{*}$ and ${e_{1}},\ldots,{e_{n}}\geq 1$ appearing in the polynomial ${a_{i}}$, is odd. This completes the proof. ∎

Corollary 1**.**

If $q$ is odd or $q=2^{m}$, $m\geq 2$ then we have ${\pi}_{q}\left({\mathop{\mathrm{EA}}}_{n}\left(\mathbb{F}_{q}\right)\right)\subset\mathop{\mathrm{Alt}}({\mathbb{F}_{q}^{n}})$.

Remark 1**.**

Let ${\theta}:\mathbb{F}_{q}\rightarrow\mathbb{C}$ be a map satisfying that ${\theta}\left(c\right)=0$ when $c=0$, and ${\theta}\left(c\right)=1$ when $c\in\mathbb{F}_{q}^{*}$. Then one can also prove Main Theorem 1 by replacing ${\chi}\left({c_{i}}\right)^{\ell}$ with ${\theta}\left({c_{i}}\right)$ in the proof of Main Theorem 1. **

Remark 2**.**

Note that Main Theorem 1 is similar to [5, Lemma 6.4], but Main Theorem 1 is more general result than [5, Lemma 6.4]. **

Example 1**.**

Let ${a_{i}}:={\alpha}{X_{1}}\cdots{X_{i-1}}{X_{i+1}}\cdots{X_{n}},{b_{i}}:={\beta}{X_{1}^{2}}{X_{2}}\cdots{X_{i-1}}{X_{i+1}}\cdots{X_{n}}\in\mathbb{F}_{q}[{X_{1}},\ldots,\hat{X_{i}},\ldots,{X_{n}}]$, and ${\alpha},{\beta}\in\mathbb{F}_{q}^{*}$. We consider the sign of the permutations induced by the following elementary automorphisms:

[TABLE]

[TABLE]

and

[TABLE]

We first assume that $q$ is odd or $q=2^{m}$, $m\geq 2$. Then by Main Theorem 1, we have $\mathop{\mathrm{sgn}}\left({\pi}_{q}\left(E_{a_{i}}^{(q)}\right)\right)=\mathop{\mathrm{sgn}}\left({\pi}_{q}\left(E_{b_{i}}^{(q)}\right)\right)=1$. By the fact that ${\pi}_{q}$ and $\mathop{\mathrm{sgn}}$ are group homomorphisms, we also have $\mathop{\mathrm{sgn}}\left({\pi}_{q}\left(E_{a_{i}+b_{i}}^{(q)}\right)\right)=\mathop{\mathrm{sgn}}\left({\pi}_{q}\left(E_{a_{i}}^{(q)}\right)\right)\times\mathop{\mathrm{sgn}}\left({\pi}_{q}\left(E_{b_{i}}^{(q)}\right)\right)=1$. We next suppose that $q=2$. Since ${\alpha},{\beta}\in\mathbb{F}_{q}^{*}$, we remark that ${\alpha}={\beta}=1$. From Main Theorem 1, we have $\mathop{\mathrm{sgn}}\left({\pi}_{q}\left(E_{a_{i}}^{(q)}\right)\right)=\mathop{\mathrm{sgn}}\left({\pi}_{q}\left(E_{b_{i}}^{(q)}\right)\right)=-1$. Again, by the fact that ${\pi}_{q}$ and $\mathop{\mathrm{sgn}}$ are group homomorphisms, we obtain $\mathop{\mathrm{sgn}}\left({\pi}_{q}\left(E_{a_{i}+b_{i}}^{(q)}\right)\right)=\mathop{\mathrm{sgn}}\left({\pi}_{q}\left(E_{a_{i}}^{(q)}\right)\right)\times\mathop{\mathrm{sgn}}\left({\pi}_{q}\left(E_{b_{i}}^{(q)}\right)\right)=1$. We can directly prove that $\mathop{\mathrm{sgn}}\left({\pi}_{q}\left(E_{a_{i}+b_{i}}^{(q)}\right)\right)=1$ by using the fact that ${\pi}_{q}\left(E_{a_{i}+b_{i}}^{(q)}\right)={\pi}_{q}\left(\left({X_{1}},\ldots,{X_{n}}\right)\right)$. **

4. Sign of permutations induced by affine automorphisms

In this section, we consider the sign of permutations induced by affine automorphisms over finite fields. Suppose that $\widetilde{A_{\mathbf{b}}^{(q)}}$ is an affine automorphism over a finite field, where,

[TABLE]

and $a_{i,j},{b_{i}}\in\mathbb{F}_{q}$ for $1\leq i,j\leq n$. We also assume that $A^{(q)}$ is the homogeneous part (linear automorphism) of the affine automorphism (4.1), namely,

[TABLE]

By Equation (2.1) and by Main Theorem 1, we obtain

[TABLE]

Thus, it is sufficient to consider the sign of affine automorphisms over finite field of the form (4.2). We put

[TABLE]

[TABLE]

and

[TABLE]

where $c\in\mathbb{F}_{q}^{*}$. It is easy to see that

[TABLE]

[TABLE]

and

[TABLE]

Since each invertible matrix is a product of elementary matrices ([1, Proposition 2.18]), a linear automorphism can be written as a finite composition of linear automorphisms of form (4.4), (4.5), and (4.6). Namely, there exists $\ell_{A}\in\mathbb{Z}_{\geq 0}$ and linear automorphisms ${M_{1}^{(A)}},\ldots,{M_{\ell_{A}}^{(A)}}$ such that

[TABLE]

where ${M_{i}^{(A)}}$ is a linear automorphism of form (4.4), (4.5), or (4.6) for each $i$ ($1\leq i\leq{\ell_{A}}$). We remark that the representation (4.7) is not unique in general. Since ${\pi}_{q}$ and $\mathop{\mathrm{sgn}}$ are group homomorphisms, we obtain

[TABLE]

Therefore, it is sufficient to consider the sign of the permutation induced by a linear automorphism of form (4.4), (4.5), and (4.6). We use for the symbols ${N_{T}}\left(A\right)$, ${N_{D}}\left(A\right)$, and ${N_{R}}\left(A\right)$ to represent the number of linear automorphisms of form (4.4), (4.5), and (4.6) appearing in (4.7), respectively. Then we have

[TABLE]

Furthermore, we suppose that $\left\{i_{1}^{(A)},\ldots,i_{{N_{D}}(A)}^{(A)}\right\}$ is the subset of $\left\{1,\ldots,{\ell_{A}}\right\}$ satisfying the following two conditions:

(i): There exist ${u_{j}^{(A)}}\in\{1,\ldots,n\}$ and $c_{j}^{(A)}\in\mathbb{F}_{q}^{*}$ such that ${M_{i_{j}^{(A)}}^{(A)}}=D_{u_{j}^{(A)}}\left(c_{j}^{(A)}\right)$ for $1\leq j\leq{N_{D}}(A)$.

(ii): For $i\in\left\{1,\ldots,{\ell_{A}}\right\}\setminus\{i_{1}^{(A)},\ldots,i_{{N_{D}}(A)}^{(A)}\}$, ${M_{i}^{(A)}}$ is of form (4.4) or (4.6).

In the following, we determine the sign of permutations induced by the above three types of linear automorphisms of form (4.4), (4.5), and (4.6).

Lemma 1**.**

(Sign of ${\pi_{q}}\left({T_{i,j}}\right)$)* Suppose that ${T_{i,j}}\in{\mathop{\mathrm{Aff}}}_{n}\left(\mathbb{F}_{q}\right)$ is a linear automorphism of form (4.4). If $n\geq 2$, then*

[TABLE]

Proof.

Put

[TABLE]

Since

[TABLE]

we have $\sharp\mathcal{B}\left({T_{i,j}}\right)=q^{n}-q^{n-2}\times q=q^{n-1}\left(q-1\right)$. Hence one can see that the permutation ${\pi_{q}}\left({T_{i,j}}\right)$ is a compositon of $q^{n-1}\left(q-1\right)/2$ transpositions on $\mathbb{F}_{q}^{n}$. We write ${N_{T_{i,j}}}=q^{n-1}\left(q-1\right)/2$.

**Case 1. ** $q={2^{m}},m\geq 2$.

Since $q^{n-1}/2=2^{m\left(n-1\right)-1}\geq 2$, we have ${N_{T_{i,j}}}\equiv 0\bmod 2$. Therefore, ${\pi_{q}}\left({T_{i,j}}\right)$ is an even permutation.

**Case 2. ** $q=2$.

In this case, we see that ${N_{T_{i,j}}}=2^{n-2}$. If $n\geq 3$, we obtain ${N_{T_{i,j}}}\equiv 0\bmod 2$. Otherwise ${N_{T_{i,j}}}=1$.

**Case 3. ** $q$ is odd.

If $q$ is odd then $q\equiv 1\bmod 4$ or $q\equiv 3\bmod 4$. From this fact, we obtain

[TABLE]

Thus, Equation (4.9) holds. ∎

Lemma 2**.**

(Sign of ${\pi_{q}}\left({D_{i}}\left(c\right)\right)$)* Suppose that ${D_{i}}\left(c\right)\in{\mathop{\mathrm{Aff}}}_{n}\left(\mathbb{F}_{q}\right)$ is a linear automorphism of form (4.5). If $n\geq 2$, then*

[TABLE]

Proof.

Let us suppose that $q$ is even. We assume that $\mathop{\mathrm{sgn}}\left({\pi_{q}}\left({D_{i}}\left(c\right)\right)\right)=-1$. Since $\mathop{\mathrm{sgn}}$ and ${\pi_{q}}$ are group homomorphisms, we have

[TABLE]

On the other hand, from ${D_{i}}\left(c\right)^{q-1}=\left({X_{1}},\ldots,{X_{n}}\right)$, we must have

[TABLE]

This is a contradiction. Therefore, $\mathop{\mathrm{sgn}}\left({\pi_{q}}\left({D_{i}}\left(c\right)\right)\right)=1$.

Next suppose that $q$ is odd. We define the map $H_{c}:\mathbb{F}_{q}\rightarrow\mathbb{F}_{q}$ as follows:

[TABLE]

The map $H_{c}$ is bijective, and the inverse map is $H_{c^{-1}}$. Since we can regard the map $H_{c}$ as a permutation on $\mathbb{F}_{q}$, it is obious that

[TABLE]

Let $g$ be a generator of the multiplicative group $\mathbb{F}_{q}^{*}$. We put $c=g^{h}$, $0\leq h=\text{ord}_{\mathbb{F}_{q}^{*}}\left(c\right)\leq q-2$. If $h=0$ then $\mathop{\mathrm{sgn}}\left(H_{c}\right)=1$. We assume that $h\neq 0$. If $h=1$ then the map $H_{c}$ is the length $q-1$ cycle $\left(g^{0}\;g^{1}\;\cdots\;g^{q-2}\right)$ as a permutation on $\mathbb{F}_{q}$, and hence $H_{c}=\left(g^{q-3}\;g^{q-2}\right)\circ\left(g^{q-4}\;g^{q-3}\right)\circ\cdots\circ\left(g^{1}\;g^{2}\right)\circ\left(g^{0}\;g^{1}\right)$, the product of $q-2$ transpositions as a permutation on $\mathbb{F}_{q}$. This yields that for $1\leq h\leq q-2$, the map $H_{c}$ is the product of $h$ copies of the length $q-1$ cycle $\left(g^{0}\;g^{1}\;\cdots\;g^{q-2}\right)$, namely, the product of $h\times\left(q-2\right)$ transpositions as a permutation on $\mathbb{F}_{q}$. Therefore, we obtain $\mathop{\mathrm{sgn}}\left(H_{c}\right)=\left(-1\right)^{h\left(q-2\right)}$. Since $q$ is odd, it satisfies that $\left(-1\right)^{h\left(q-2\right)}=\left(\left(-1\right)^{q-2}\right)^{h}=\left(-1\right)^{h}$. Hence

[TABLE]

for $c\in\mathbb{F}_{q}^{*}$, $c\neq 1$. Equation (4.11) is obviously true for $c=1$. Thus the assertion holds. ∎

Lemma 3**.**

(Sign of ${\pi_{q}}\left({R_{i,j}}\left(c\right)\right)$)* Suppose that ${R_{i,j}}\left(c\right)\in{\mathop{\mathrm{Aff}}}_{n}\left(\mathbb{F}_{q}\right)$ is a linear automorphism of form (4.6). If $n\geq 2$, then*

[TABLE]

Proof.

By ${R_{i,j}}\left(c\right)\in{\mathop{\mathrm{Aff}}}_{n}\left(\mathbb{F}_{q}\right)\cap{\mathop{\mathrm{EA}}}_{n}\left(\mathbb{F}_{q}\right)\subset{\mathop{\mathrm{EA}}}_{n}\left(\mathbb{F}_{q}\right)$, it follows immediately from Main Theorem 1. ∎

We are now in the position to prove the main result of this section.

Main Theorem 2**.**

(Sign of affine automorphisms)  With notation as above, the following assertions are hold:

(1)* If $q={2^{m}}$, $m\geq 2$ then*

[TABLE]

(2)* If $q=2$ and $n=2$ then*

[TABLE]

(3)* If $q=2$ and $n\geq 3$ then*

[TABLE]

(4)* If $q$ is odd then*

[TABLE]

In particular, if $q\equiv 1\bmod 4$ then

[TABLE]

Proof.

The assertions (1) through (4) follow immediately from Equation (4.3), Equation (4.7), and Lemma 1 through Lemma 3. This completes the proof. ∎

Corollary 2**.**

If $q=2^{m}$ and $m\geq 2$, or $q=2$ and $n\geq 3$ then we have ${\pi}_{q}\left({\mathop{\mathrm{Aff}}}_{n}\left(\mathbb{F}_{q}\right)\right)\subset\mathop{\mathrm{Alt}}({\mathbb{F}_{q}^{n}})$.

Remark 3**.**

One can see that Equation (4.14), (4.15), (4.16), and (4.17) depend on the representation (4.7) which is not unique. However, since ${\pi}_{q}\left(\widetilde{A_{\mathbf{b}}^{(q)}}\right)$ is uniquely determined as a permutation on $\mathbb{F}_{q}^{n}$, $\mathop{\mathrm{sgn}}\left({\pi}_{q}\left(\widetilde{A_{\mathbf{b}}^{(q)}}\right)\right)$ does not depend on the representation (4.7). **

Example 2**.**

Let ${\alpha},{\beta}\in\mathbb{F}_{q}^{*}$. We consider the sign of the permutation induced by the affine automorphism $A^{(q)}:=\left({X_{3}},{X_{2}},{\alpha}{X_{1}}+{\beta}{X_{3}}\right)\in{\mathop{\mathrm{Aff}}}_{3}\left(\mathbb{F}_{q}\right)$. It is easy to see that $A^{(q)}=\left({X_{3}},{X_{2}},{X_{1}}\right)\circ\left({X_{1}}+{\beta}{X_{3}},{X_{2}},{X_{3}}\right)\circ\left({\alpha}{X_{1}},{X_{2}},{X_{3}}\right)=T_{1,3}\circ R_{1,3}\left({\beta}\right)\circ D_{1}\left({\alpha}\right)$. We remark that ${N_{T}}\left(A\right)={N_{D}}\left(A\right)={N_{R}}\left(A\right)=1$, ${\ell_{A}}=3$, and $\mathop{\mathrm{sgn}}\left({\pi}_{q}\left(A^{(q)}\right)\right)=\mathop{\mathrm{sgn}}\left({\pi}_{q}\left(T_{1,3}\right)\right)\times\mathop{\mathrm{sgn}}\left({\pi}_{q}\left(R_{1,3}\left({\beta}\right)\right)\right)\times\mathop{\mathrm{sgn}}\left({\pi}_{q}\left(D_{1}\left({\alpha}\right)\right)\right)$. If $p=2$ then by Equation (4.13) and by Equation (4.15), one can easily see that $\mathop{\mathrm{sgn}}\left({\pi}_{q}\left(A^{(q)}\right)\right)=1$. If $q$ is odd then by Equation (4.16),

[TABLE]

In particular, if $q\equiv 1\bmod 4$ then $\mathop{\mathrm{sgn}}\left({\pi}_{q}\left(A^{(q)}\right)\right)=\left(-1\right)^{{\rm ord}_{\mathbb{F}_{q}^{*}}\left({\alpha}\right)}$, and if $q\equiv 3\bmod 4$ then $\mathop{\mathrm{sgn}}\left({\pi}_{q}\left(A^{(q)}\right)\right)=-1\times\left(-1\right)^{{\rm ord}_{\mathbb{F}_{q}^{*}}\left({\alpha}\right)}=\left(-1\right)^{{\rm ord}_{\mathbb{F}_{q}^{*}}\left({\alpha}\right)+1}$. **

5. Sign of permutations induced by triangular automorphisms and tame automorphisms

In this section, we consider the sign of permutations induced by triangular automorphisms and tame automorphisms over finite fields. By Main Theorem 1 and Main Theorem 2, we obtain the following corollary (Corollary 3).

Corollary 3**.**

(Sign of triangular automorphisms)  Suppose that $J_{a,f}^{(q)}$ is a triangular automorphism over a finite field, namely,

[TABLE]

If $q=2^{m}$, $m\geq 2$ then ${\pi}_{q}\left(J_{a,f}^{(q)}\right)\in\mathop{\mathrm{Alt}}({\mathbb{F}_{q}^{n}})$. Namely, if $q=2^{m}$, $m\geq 2$ then

[TABLE]

If $q$ is odd then

[TABLE]

In other words, if $q$ is odd then $\mathop{\mathrm{sgn}}\left({\pi}_{q}\left(J_{a,f}^{(q)}\right)\right)$ depends only on the coefficients ${a_{1}},\ldots,{a_{n}}$. If $q=2$ then

[TABLE]

where $M_{f_{1}}$ is the number of monomials of the form $cX_{2}^{e_{2}}{\cdots}X_{n}^{e_{n}}$ with $c\in\mathbb{F}_{q}^{*}$ and ${e_{2}},\ldots,{e_{n}}\geq 1$ appearing in the polynomial ${f_{1}}\in\mathbb{F}_{q}[{X_{2}},\ldots,{X_{n}}]$.

Proof.

By using the notation of Equation (3.1) and Equation (4.5), we have

[TABLE]

Hence we obtain

[TABLE]

By Equation (5.5), Main Theorem 1, and Main Theorem 2, we obtain the desired results. ∎

Corollary 4**.**

(Sign of strictly triangular automorphisms)  Suppose that $J_{f}^{(q)}$ is a strictly triangular automorphism over a finite field, namely,

[TABLE]

If $q$ is odd or $q=2^{m}$, $m\geq 2$ then ${\pi}_{q}\left(J_{f}^{(q)}\right)\in\mathop{\mathrm{Alt}}({\mathbb{F}_{q}^{n}})$. Namely, if $q$ is odd or $q=2^{m}$, $m\geq 2$ then

[TABLE]

If $q=2$ then

[TABLE]

where $M_{f_{1}}$ is the number of monomials of the form $cX_{2}^{e_{2}}{\cdots}X_{n}^{e_{n}}$ with $c\in\mathbb{F}_{q}^{*}$ and ${e_{2}},\ldots,{e_{n}}\geq 1$ appearing in the polynomial ${f_{1}}\in\mathbb{F}_{q}[{X_{2}},\ldots,{X_{n}}]$.

Proof.

It follows immediately from Corollary 3. ∎

We recall that for any ${\phi}^{(q)}\in\mathop{\mathrm{TA}_{n}}(\mathbb{F}_{q})$, there exist $l\in\mathbb{Z}_{\geq 0}$, ${\epsilon_{1}},{\epsilon_{2}}\in\{0,1\}\subset\mathbb{Z}$, $\widetilde{A_{s,\mathbf{b}^{(s)}}^{(q)}}\in\mathop{\mathrm{Aff}}_{n}(\mathbb{F}_{q})$ ($1\leq s\leq l+1$) of the form

[TABLE]

and $J_{s,{t^{(s)}},{f^{(s)}}}^{(q)}\in\mathop{\mathrm{BA}_{n}}(\mathbb{F}_{q})$ ($1\leq s\leq l$) of the form

[TABLE]

such that

[TABLE]

$\widetilde{A_{s,\mathbf{b}^{(s)}}^{(q)}}\not\in\mathop{\mathrm{BA}_{n}}(\mathbb{F}_{q})$ for $2\leq s\leq l+1$, and $J_{s,{t^{(s)}},{f^{(s)}}}^{(q)}\not\in\mathop{\mathrm{Aff}}_{n}(\mathbb{F}_{q})$ for $1\leq s\leq l$ (for example, [2, Lemma 5.1.1]).

We use the symbol $A_{s}^{(q)}$ to denote the homogeneous part (linear automorphism) of the affine automorphism $\widetilde{A_{s,\mathbf{b}^{(s)}}^{(q)}}$ (as in Equation (4.1) and Equation (4.2)) for $1\leq s\leq l+1$. Furthermore, for each $s$ ($1\leq s\leq l$), we denote by $M_{f_{1}^{(s)}}$ the number of monomials of the form $cX_{2}^{e_{2}}{\cdots}X_{n}^{e_{n}}$ with $c\in\mathbb{F}_{q}^{*}$ and ${e_{2}},\ldots,{e_{n}}\geq 1$ appearing in the polynomial ${f_{1}^{(s)}}\in\mathbb{F}_{q}[{X_{2}},\ldots,{X_{n}}]$.

The following corollary (Corollary 5) states that if we know Equation (5.9) for a given ${\phi}^{(q)}\in\mathop{\mathrm{TA}_{n}}(\mathbb{F}_{q})$, then one can easily compute the sign of the permutation induced by ${\phi}^{(q)}\in\mathop{\mathrm{TA}_{n}}(\mathbb{F}_{q})$.

Corollary 5**.**

(Sign of tame automorphisms)  With notation as above, the following assertions are hold:

(1)* If $q={2^{m}}$, $m\geq 2$ then*

[TABLE]

(2)* If $q=2$ and $n=2$ then*

[TABLE]

(3)* If $q=2$ and $n\geq 3$ then*

[TABLE]

(4)* If $q$ is odd then*

[TABLE]

In particular, if $q\equiv 1\bmod 4$ then

[TABLE]

Proof.

By Equation (5.9), we obtain

[TABLE]

By Equation (5.15), Main Theorem 2, and Corollary 3, we obtain the desired results. ∎

Remark 4**.**

As in Remark 3, one can see that Equation (5.11), (5.12), (5.13), and (5.14) depend on the representations (4.7) and (5.9) which are not unique. However, since ${\pi}_{q}\left({\phi}^{(q)}\right)$ is uniquely determined as a permutation on $\mathbb{F}_{q}^{n}$, $\mathop{\mathrm{sgn}}\left({\pi}_{q}\left({\phi}^{(q)}\right)\right)$ does not depend on the representations (4.7) and (5.9). **

Remark 5**.**

Suppose that $n$ is greater than or equal to two. Corollary 5 tells us that ${\pi}_{q}\left(\mathop{\mathrm{TA}_{n}}\left(\mathbb{F}_{q}\right)\right)\subset\mathop{\mathrm{Sym}}\left({\mathbb{F}_{q}^{n}}\right)$ (this is a trivial inclusion relation) if $q$ is odd or $q=2$, and ${\pi}_{q}\left(\mathop{\mathrm{TA}_{n}}\left(\mathbb{F}_{q}\right)\right)\subset\mathop{\mathrm{Alt}}\left({\mathbb{F}_{q}^{n}}\right)$ if $q={2^{m}}$ and $m\geq 2$. This indicates that Corollary 5 is strictly weaker than [3, Theorem 2.3]. However, Main Theorem 1, Main Theorem 2, Corollary 3, and Corollary 5 are useful for determining the sign of permutations induced by tame automorphisms over finite fields. The reasons are as follows. Firstly, for $q={2^{m}}$ and $m\geq 2$, we prove that ${\pi}_{q}\left(\mathop{\mathrm{TA}_{n}}(\mathbb{F}_{q})\right)\subset\mathop{\mathrm{Alt}}({\mathbb{F}_{q}^{n}})$ by directly showing that ${\pi}_{q}\left({\mathop{\mathrm{EA}}}_{n}\left(\mathbb{F}_{q}\right)\right)\subset\mathop{\mathrm{Alt}}({\mathbb{F}_{q}^{n}})$ and ${\pi}_{q}\left({\mathop{\mathrm{Aff}}}_{n}\left(\mathbb{F}_{q}\right)\right)\subset\mathop{\mathrm{Alt}}({\mathbb{F}_{q}^{n}})$. Secondly, if $q$ is odd then one can not determine the sign of permutation induced by an elementary automorphisms over a finite field by using [3, Theorem 2.3]. In contrast to [3, Theorem 2.3], Main Theorem 1 tells us that if $q$ is odd then each permutation induced by an elementary automorphism over a finite field is even. In other words, if $q$ is odd then ${\pi}_{q}\left({\mathop{\mathrm{EA}}}_{n}\left(\mathbb{F}_{q}\right)\right)\subset\mathop{\mathrm{Alt}}({\mathbb{F}_{q}^{n}})$ (Corollary 1). Similarly, in contrast to [3, Theorem 2.3], Main Theorem 2 tells us that if $q=2$ and $n\geq 3$ then each permutation induced by an affine automorphism over a finite field is even. Namely, if $q=2$ and $n\geq 3$ then ${\pi}_{q}\left({\mathop{\mathrm{Aff}}}_{n}\left(\mathbb{F}_{q}\right)\right)\subset\mathop{\mathrm{Alt}}({\mathbb{F}_{q}^{n}})$ (Corollary 2). Thus, our results (Main Theorem 1, Main Theorem 2, Corollary 3, and Corollary 5) and [3, Theorem 2.3] are complementary to each other. **

Acknowledgements

This work was supported by JSPS KAKENHI Grant-in-Aid for Young Scientists (B) 16K16066.