Classification of multivariate skew polynomial rings over finite fields via affine transformations of variables

TL;DR

This paper classifies multivariate skew polynomial rings over finite fields by showing they are all isomorphic to a simplified form via affine transformations, with explicit computational methods provided.

Contribution

It provides a complete classification of multivariate skew polynomial rings over finite fields using affine transformations and diagonalization of ring morphisms.

Findings

All ring morphisms from the field to matrices are diagonalizable.

Multivariate skew polynomial rings over finite fields are isomorphic to a simplified diagonal form.

The simplified form can be computed explicitly and efficiently.

Abstract

In this work, free multivariate skew polynomial rings are considered, together with their quotients over ideals of skew polynomials that vanish at every point (which includes minimal multivariate skew polynomial rings). We provide a full classification of such multivariate skew polynomial rings (free or not) over finite fields. To that end, we first show that all ring morphisms from the field to the ring of square matrices are diagonalizable, and that the corresponding derivations are all inner derivations. Secondly, we show that all such multivariate skew polynomial rings over finite fields are isomorphic as algebras to a multivariate skew polynomial ring whose ring morphism from the field to the ring of square matrices is diagonal, and whose derivation is the zero derivation. Furthermore, we prove that two such representations only differ in a permutation of the field automorphisms appearing in the corresponding diagonal. The algebra isomorphisms are given by affine transformations of variables and preserve evaluations and degrees. In addition, ours proofs show that the simplified form of multivariate skew polynomial rings can be found computationally and explicitly.

Equations

$$F(\mathbf{x})=\sum _{\mathfrak{m}(\mathbf{x})\in \mathcal{M}}{F}_{\mathfrak{m}}\mathfrak{m}(\mathbf{x}),$$

$$x_{i}a=\sum _{j=1}^n{\sigma }_{i,j}(a)x_j+{\delta }_i(a),$$

$$\sigma:\mathbb{F}\longrightarrow\mathbb{F}^{n\times n}:a\mapsto\left(\begin{array}[]{cccc}\sigma_{1,1}(a)&\sigma_{1,2}(a)&\ldots&\sigma_{1,n}(a)\\ \sigma_{2,1}(a)&\sigma_{2,2}(a)&\ldots&\sigma_{2,n}(a)\\ \vdots&\vdots&\ddots&\vdots\\ \sigma_{n,1}(a)&\sigma_{n,2}(a)&\ldots&\sigma_{n,n}(a)\\ \end{array}\right)$$

$$\delta:\mathbb{F}\longrightarrow\mathbb{F}^{n}:a\mapsto\left(\begin{array}[]{c}\delta_{1}(a)\\ \delta_{2}(a)\\ \vdots\\ \delta_{n}(a)\end{array}\right)$$

$$\delta (ab)=\sigma (a)\delta (b)+\delta (a)b,$$

$$\mathbf{x}a=\sigma (a)\mathbf{x}+\delta (a).$$

$$F(\mathbf{x})=\sum _{i=1}^n{G}_i(\mathbf{x})(x_i-a_i)+b.$$

$$F(\mathbf{a})=E_{\mathbf{a}}^{\sigma ,\delta }(F(\mathbf{x}))\in \mathbb{F}.$$

$$I({\mathbb{F}}_q^n)=\left(\{x_i{x}_j-x_j{x}_i\mid 1\le i<j\le n\}\cup \{x_1^q-x_1,x_2^q-x_2,\dots ,x_n^q-x_n\}\right).$$

$$x_{i}a={\sigma }_i(a)x_i,$$

$$\delta (a)=\lambda (a-\sigma (a)),$$

$$\delta (ab)=\sigma (a)\delta (b)+\tau (b)\delta (a),$$

$$\delta (a)=(\tau (a)-\sigma (a))\mathbf{\lambda },$$

$$\delta \left(a^{j+1}\right)=\left(\sum _{i=0}^j\sigma (a{)}^i\tau (a{)}^{j-i}\right)\delta (a),$$

$$\left(\genfrac{}{}{0ex}{}{q-1}{i}\right)=(-1{)}^i$$

$$(x-y)(x-y{)}^{q-1}=(x-y{)}^q=x^q-y^q=(x-y)\left(\sum _{i=0}^{q-1}{x}^i{y}^{q-1-i}\right).$$

$$\sum _{i=0}^{q-1}(-1{)}^i(\genfrac{}{}{0ex}{}{q-1}{i})x^i{y}^{q-1-i}=(x-y{)}^{q-1}=\sum _{i=0}^{q-1}{x}^i{y}^{q-1-i},$$

$$(T-S{)}^{q-1}=\sum _{i=0}^{q-1}(-1{)}^i\left(\genfrac{}{}{0ex}{}{q-1}{i}\right)T^i{S}^{q-1-i}=\sum _{i=0}^{q-1}{T}^i{S}^{q-1-i}.$$

$$\delta (c)=\delta (c^q)=\left(\sum _{i=0}^{q-1}{T}^i{S}^{q-1-i}\right)\delta (c)=(T-S{)}^{q-1}\delta (c).$$

$$\mathbf{\lambda }=(T-S{)}^{q-2}\delta (c)\in {\mathbb{F}}_q^n.$$

$$\delta (c)=(\tau (c)-\sigma (c))\mathbf{\lambda }.$$

$$\partial (a)=(\tau (a)-\sigma (a))\mathbf{\lambda },$$

$$\delta (a)=\delta \left(c^{j+1}\right)=\left(\sum _{i=0}^j\tau (c{)}^i\sigma (c{)}^{j-i}\right)\delta (c)=\left(\sum _{i=0}^j\tau (c{)}^i\sigma (c{)}^{j-i}\right)\partial (c)=\partial (a),$$

$$\sigma(f(z))=\left(\begin{array}[]{cc}f(z)&f^{\prime}(z)\\ 0&f(z)\end{array}\right),$$

$$\delta(f(z))=\left(\begin{array}[]{c}f^{\prime\prime}(z)\\ 2f^{\prime}(z)\end{array}\right),$$

$$\sigma(a)=A{\rm diag}(\sigma_{1}(a),\sigma_{2}(a),\ldots,\sigma_{n}(a))A^{-1}=A\left(\begin{array}[]{cccc}\sigma_{1}(a)&0&\ldots&0\\ 0&\sigma_{2}(a)&\ldots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\ldots&\sigma_{n}(a)\\ \end{array}\right)A^{-1},$$

$$S=A\left(\begin{array}[]{cccc}\Lambda_{1}&0&\ldots&0\\ 0&\Lambda_{2}&\ldots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\ldots&\Lambda_{n}\\ \end{array}\right)A^{-1},$$

$$\Lambda_{i}=\left(\begin{array}[]{cccccc}\lambda_{i}&\varepsilon_{i,1}&0&\ldots&0&0\\ 0&\lambda_{i}&\varepsilon_{i,2}&\ldots&0&0\\ 0&0&\lambda_{i}&\ldots&0&0\\ \vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\ 0&0&0&\ldots&\lambda_{i}&\varepsilon_{i,n_{i}-1}\\ 0&0&0&\ldots&0&\lambda_{i}\\ \end{array}\right)\in\mathbb{F}_{q^{r}}^{n_{i}\times n_{i}},$$

$$\sigma(a)=A\left(\begin{array}[]{cccc}\tau_{1}(a)&0&\ldots&0\\ 0&\tau_{2}(a)&\ldots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\ldots&\tau_{k}(a)\\ \end{array}\right)A^{-1},$$

$$\tau_{i}(a)=\left(\begin{array}[]{cccc}\varphi^{(i)}(a)&\varphi_{1,2}^{(i)}(a)&\ldots&\varphi_{1,n_{i}}^{(i)}(a)\\ 0&\varphi^{(i)}(a)&\ldots&\varphi_{2,n_{i}}^{(i)}(a)\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\ldots&\varphi^{(i)}(a)\\ \end{array}\right),$$

Full text

Keywords: Affine transformations, derivations, free polynomial rings, Moore matrices, multivariate skew polynomial rings, Vandermonde matrices.

MSC: 11T06, 11T30, 12E10, 12E20.

1 Introduction

Univariate skew polynomial rings over a division ring $\mathbb{F}$ (or any ring in general) were formally introduced in [22] and further studied over finite fields $\mathbb{F}_{q}$ in [21], although some explicit applications of this latter case can be found much earlier [18]. These rings are “non-commutative polynomial rings” $\mathcal{R}$ in one variable $x$ (formally, $\mathcal{R}$ is a left vector space over $\mathbb{F}$ with infinite left basis $\{x^{i}\mid i=0,1,2,\ldots\}$) such that addition is as usual and multiplication satisfies the following: $\mathcal{R}$ is a (commutative or non-commutative) ring with identity $1=x^{0}$, where the $i$th power of $x=x^{1}$ corresponds to the monomial $x^{i}$ and the degree of a product of two polynomials is the sum of their degrees.

Such algebraic structures on the left vector space $\mathcal{R}$ with left basis $\{x^{i}\mid i=0,1,2,\ldots\}$ correspond bijectively with pairs $(\sigma,\delta)$, where $\sigma:\mathbb{F}\longrightarrow\mathbb{F}$ is a ring endomorphism and $\delta:\mathbb{F}\longrightarrow\mathbb{F}$ is a $\sigma$-derivation. In other words, $\delta$ is additive and, for all $a,b\in\mathbb{F}$, it holds that $\delta(ab)=\sigma(a)\delta(b)+\delta(a)b$, for all $a,b\in\mathbb{F}$. This bijection is due to the relation $xa=\sigma(a)x+\delta(a)$, for all $a\in\mathbb{F}$, that needs to happen in $\mathcal{R}$ in order to satisfy the properties mentioned above. See [22, pp. 481–482] for more details.

Since univariate skew polynomial rings are right Euclidean domains [22, p. 483], the evaluation of a skew polynomial $F(x)\in\mathcal{R}$ over a point $a\in\mathbb{F}$ can be defined as the remainder of the Euclidean division of $F(x)$ by $x-a$ on the right. This concept of evaluation was first considered in full generality by Lam and Leroy in [8, 9]. Among others, this notion of evaluation helps unify the study of Vandermonde, Moore and Wronskian matrices [8, 9] and more general matrix types (see [15, p. 604] for instance), and gives a natural framework for Hilbert 90 Theorems [10] and pseudolinear transformations [12]. Specifically over finite fields, this notion has provided maximum rank distance (error-correcting) codes [6] (based on Moore matrices [18]), and maximum sum-rank distance codes [15] with finite-field sizes that are not exponential in the code length, in constrast with [6] (see [15, Sec. 4.2]).

Recently, two natural extensions of this concept of evaluation have been given for multivariate skew polynomials. In [7], it is proposed to evaluate certain iterated skew polynomials [5, Sec. 8.8] over certain affine points by a multivariate Euclidean division algorithm. This lack of universality is due to the lack of unique remainders when dividing an iterated skew polynomial by linear skew polynomials $x_{1}-a_{1}$, $x_{2}-a_{2}$, $\ldots$, $x_{n}-a_{n}$, which is due to the relations between variables in iterated skew polynomial rings with $n>1$ variables (see [7, Ex. 3.5] and [16, Remarks 7 & 8]). Alternatively, a natural and universal definition of evaluation was given in [16, Def. 9] for free multivariate skew polynomials, precisely due to the lack of relations among the variables. Due to [16, Lemma 5], any free multivariate skew polynomial can be evaluated over any affine point as the unique remainder of the division by $x_{1}-a_{1}$, $x_{2}-a_{2}$, $\ldots$, $x_{n}-a_{n}$ on the right. Multivariate skew polynomial rings with relations on the variables and where evaluation is still universal can then by defined as quotients of the free ring by two-sided ideals of skew polynomials that vanish at every point [16, Def. 19]. However, these rings are not iterated skew polynomial rings in general.

Thanks to the definitions considered in [16], the central results from [8, 9] on the roots of skew polynomials were translated from the univariate to the multivariate case in [16]. Among these results, the matroidal or lattice structure of the sets of roots of univariate skew polynomials played a crucial role in the properties of the matrices of Vandermonde type considered in [8, 9, 15] and of the linear codes in [6, 15], especially when considered over finite fields. Apart from its own interest, one of the main motivations behind [7, 16] is to define linear codes of Reed-Muller type [19, 23] in the form of evaluation codes that would share the features of the codes in [6, 15], but allowing more general parameters and properties. As a collateral consequence, it was recently shown in [14] that Hilbert’s Theorem 90 over general Galois extensions of fields (as proven originally by Noether [20], see also [1, Th. 21]) can be obtained from such a structure of the roots of multivariate skew polynomials.

To understand the extent of such applications, it is of interest to classify multivariate skew polynomial rings and to find explicit descriptions of the ring morphisms $\sigma:\mathbb{F}\longrightarrow\mathbb{F}^{n\times n}$ and the $\sigma$-derivations $\delta:\mathbb{F}\longrightarrow\mathbb{F}^{n}$ that define such rings (see [16, Defs. 1 & 2] or Section 2 below). In the univariate case over finite fields, it is well-known that all endomorphisms are of Frobenius type [13, Th. 2.21] and all derivations are inner derivations [5, Sec. 8.3] (see also [15, Sec. 4.2] or Propositions 1 and 2 below). To the best of our knowledge, explicit descriptions are not yet known when $n>1$.

The first main objective and results in this paper are showing that, in the multivariate case over finite fields, all morphisms are diagonalizable (Theorem 2 in Subsection 3.2) and all derivations are inner derivations (Theorem 1 in Subsection 3.1), providing explicit descriptions of such objects. As in the univariate case, we believe that such characterizations of morphisms and derivations over finite fields are of interest by themselves. Furthermore, these characterizations do not hold in general for infinite fields (see Examples 1 and 3). Moreover, the well-known Skolem-Noether theorem [24] cannot be directly applied to characterize ring morphisms $\sigma:\mathbb{F}\longrightarrow\mathbb{F}^{n\times n}$ over finite fields (see Subsection 3.2).

The second main objective and results of this paper are showing that the given characterizations imply that any multivariate skew polynomial ring over a finite field is isomorphic as an algebra to a multivariate skew polynomial ring given by a diagonal morphism $\sigma={\rm diag}(\sigma_{1},\sigma_{2},\ldots,\sigma_{n})$ and a zero derivation $\delta=0$ (Theorem 5 in Section 7). Such a ring isomorphism is given by affine transformations of variables (meaning a composition of a linear tranformation and a translation), which moreover preserve evaluations and degrees (see Sections 4 and 5 for linear transformations and translations, respectively). In addition, we show that affine transformations of variables are the only left $\mathbb{F}$-algebra isomorphisms with these properties (Theorem 4 Section 6). When $\mathbb{F}$ is finite, this implies that two multivariate skew polynomial rings given by diagonal ring morphisms $\sigma={\rm diag}(\sigma_{1},\sigma_{2},\ldots,\sigma_{n})$ and $\tau={\rm diag}(\tau_{1},\tau_{2},\ldots,\tau_{n})$ are isomorphic by such an algebra isomorphism if, and only if, $(\sigma_{1},\sigma_{2},\ldots,\sigma_{n})$ is a permutation of $(\tau_{1},\tau_{2},\ldots,\tau_{n})$ (Theorem 6 in Section 7). We believe that the study of affine transformations of variables is of interest by itself, and thus we consider general division rings $\mathbb{F}$ for such results.

In conclusion, these results provide a full and explicit classification of free and non-free multivariate skew polynomial rings over finite fields, taking evaluations and degrees into account (Section 7). Furthermore, the proofs give explicit and computational methods to find the reduced form of a multivariate skew polynomial ring.

Notation

Unless otherwise stated, $\mathbb{F}$ will denote a division ring. A field is a commutative division ring. All rings in this work will be assumed to have (multiplicative) identity and all ring morphisms are required to map identities to identities. Throughout this work, $q$ is a fixed power of a prime number $p$, and $\mathbb{F}_{q}$ denotes the finite field with $q$ elements. For positive integers $m$ and $n$, $\mathbb{F}^{m\times n}$ will denote the set of $m\times n$ matrices over $\mathbb{F}$, whose identity (if $m=n$) is denoted by $I$, and $\mathbb{F}^{n}$ will denote the set of column vectors of length $n$ over $\mathbb{F}$. That is, $\mathbb{F}^{n}=\mathbb{F}^{n\times 1}$.

2 Preliminaries: Multivariate skew polynomial rings

We will consider multivariate skew polynomial rings as in [16] for the reasons stated in the Introduction. Let $x_{1},x_{2},\ldots,x_{n}$ be pair-wise distinct letters, which we will call variables, and let $\mathcal{M}$ be the free monoid with basis $x_{1},x_{2},\ldots,x_{n}$ [5, Sec. 6.5], whose elements are called monomials and whose identity is denoted by $1$. We will denote $\mathbf{x}=(x_{1},x_{2},\ldots,x_{n})^{T}\in\mathcal{M}^{n}$ and we will typically denote monomials in such variables by $\mathfrak{m}(\mathbf{x})$, or $\mathfrak{m}$ for brevity if there is no confusion about $\mathbf{x}$. We define the degree of a given monomial $\mathfrak{m}(\mathbf{x})\in\mathcal{M}$, denoted by $\deg(\mathfrak{m}(\mathbf{x}))$, as its length as a string in $x_{1},x_{2},\ldots,x_{n}$, where $\deg(1)=0$.

Let $\mathcal{R}$ be the left vector space over $\mathbb{F}$ with basis $\mathcal{M}$. We call its elements free multivariate skew polynomials, or simply skew polynomials. They are of the form

[TABLE]

where $F_{\mathfrak{m}}\in\mathbb{F}$ are all zero except for a finite number of them. We define the degree of $F(\mathbf{x})$, denoted by $\deg(F(\mathbf{x}))$, as the maximum degree of a monomial $\deg(\mathfrak{m}(\mathbf{x}))$ such that $F_{\mathfrak{m}}\neq 0$, in the case $F\neq 0$, and we define $\deg(F(\mathbf{x}))=\infty$ if $F(\mathbf{x})=0$.

Following Ore’s line of thought [22, pp. 481–482], it was shown in [16, Th. 1] that a product in $\mathcal{R}$ turns it into a ring with multiplicative identity $1$, where products of monomials consist in appending them and where $\deg(F(\mathbf{x})G(\mathbf{x}))=\deg(F(\mathbf{x}))+\deg(G(\mathbf{x}))$, for all $F(\mathbf{x}),G(\mathbf{x})\in\mathcal{R}$, if, and only if, there exist maps $\sigma_{i,j},\delta_{i}:\mathbb{F}\longrightarrow\mathbb{F}$, for $i,j=1,2,\ldots,n$, such that

[TABLE]

for $i=1,2,\ldots,n$, and for all $a\in\mathbb{F}$, where the map

[TABLE]

is a ring morphism and the map

[TABLE]

is a $\sigma$-derivation. Recall from [16, Def. 1] that $\delta$ is a $\sigma$-derivation if it is additive and

[TABLE]

for all $a,b\in\mathbb{F}$. With this compact notation, we may rewrite Equation (1) as

[TABLE]

As in the univariate case, it was shown in [16, Th. 1] that such pairs $(\sigma,\delta)$ correspond bijectively with products in $\mathcal{R}$ satisfying the properties described above. Hence we may use the notation $\mathcal{R}=\mathbb{F}[\mathbf{x};\sigma,\delta]$ when we consider the product in $\mathcal{R}$ given by the pair $(\sigma,\delta)$. In other words, $\mathbb{F}[\mathbf{x};\sigma,\delta]=\mathbb{F}[\mathbf{x};\sigma^{\prime},\delta^{\prime}]$ if, and only if, $(\sigma,\delta)=(\sigma^{\prime},\delta^{\prime})$. However, it may hold that $(\sigma,\delta)\neq(\sigma^{\prime},\delta^{\prime})$ and at the same time $\mathbb{F}[\mathbf{x};\sigma,\delta]$ and $\mathbb{F}[\mathbf{x};\sigma^{\prime},\delta^{\prime}]$ are isomorphic as left $\mathbb{F}$-algebras in a canonical way. Observe that the conventional free multivariate polynomial ring $\mathbb{F}[\mathbf{x}]$ corresponds to $\sigma={\rm Id}$ and $\delta=0$, where ${\rm Id}(a)=aI$, for all $a\in\mathbb{F}$.

Thanks to the lack of relations among the variables, it was proven in [16, Lemma 5] that, for any $a_{1},a_{2},\ldots,a_{n}\in\mathbb{F}$ and any $F(\mathbf{x})\in\mathbb{F}[\mathbf{x};\sigma,\delta]$, there exist unique $G_{1}(\mathbf{x}),G_{2}(\mathbf{x}),$ $\ldots,$ $G_{n}(\mathbf{x})\in\mathbb{F}[\mathbf{x};\sigma,\delta]$ and $b\in\mathbb{F}$ such that

[TABLE]

Hence, we may define the $(\sigma,\delta)$-evaluation of $F(\mathbf{x})$ at $\mathbf{a}\in\mathbb{F}^{n}$ as the constant $b\in\mathbb{F}$, which we will denote by

[TABLE]

The notation $F(\mathbf{a})$ is for ease of reading and brevity, whereas $E^{\sigma,\delta}_{\mathbf{a}}(F(\mathbf{x}))$ will be necessary when we consider different pairs $(\sigma,\delta)$.

Once the natural and universal evaluation in (3) is obtained over free multivariate skew polynomial rings, we may extend it to multivariate skew polynomial rings with relations on the variables as follows. Let $I(\mathbb{F}^{n})\subseteq\mathbb{F}[\mathbf{x};\sigma,\delta]$ denote the left ideal of skew polynomials that vanish at every point. It was shown in [16, Prop. 18] that $I(\mathbb{F}^{n})$ is a two-sided ideal. Therefore, we may define non-free multivariate skew polynomial rings, where evaluation is still natural and universal as in (3), as quotient rings of the form $\mathbb{F}[\mathbf{x};\sigma,\delta]/I$, where $I\subseteq I(\mathbb{F}^{n})$ is a two-sided ideal. The skew polynomial ring $\mathbb{F}[\mathbf{x};\sigma,\delta]/I(\mathbb{F}^{n})$ can be called a minimal (multivariate) skew polynomial ring. In the conventional case $(\sigma,\delta)=({\rm Id},0)$ over a finite field $\mathbb{F}=\mathbb{F}_{q}$, we recover the minimal conventional polynomial ring due to the well-known fact that

[TABLE]

Formally, our first main results will be showing that any pair $(\sigma,\delta)$ is such that $\sigma=A{\rm diag}(\sigma_{1},\sigma_{2},\ldots,\sigma_{n})A^{-1}$ (Theorem 2) and $\delta=A(\boldsymbol{\lambda}{\rm Id}-{\rm diag}(\sigma_{1},\sigma_{2},\ldots,\sigma_{n})\boldsymbol{\lambda})$ (Theorem 1), over finite fields $\mathbb{F}=\mathbb{F}_{q}$, for an invertible matrix $A\in\mathbb{F}_{q}^{n\times n}$ and a vector $\boldsymbol{\lambda}\in\mathbb{F}_{q}^{n}$. Our second main results will be proving that in such cases, the free or non-free multivariate skew polynomial ring $\mathbb{F}_{q}[\mathbf{x};\sigma,\delta]$ is canonically isomorphic to the ring $\mathbb{F}_{q}[\mathbf{x};{\rm diag}(\sigma_{1},\sigma_{2},\ldots,\sigma_{n}),0]$ (Theorem 5). Note that in the latter skew polynomial ring, the commutation rule (1) simply reads

[TABLE]

for all $i=1,2,\ldots,n$ and all $a\in\mathbb{F}_{q}$. In addition, we show in Theorem 6 that two such skew polynomial rings $\mathbb{F}_{q}[\mathbf{x};{\rm diag}(\sigma_{1},\sigma_{2},\ldots,\sigma_{n}),0]$ and $\mathbb{F}_{q}[\mathbf{x};{\rm diag}(\tau_{1},\tau_{2},\ldots,\tau_{n}),0]$ are isomorphic as $\mathbb{F}_{q}$-algebras by an isomorphism that preserves degrees if, and only if, $(\sigma_{1},\sigma_{2},\ldots,\sigma_{n})$ is a permutation of $(\tau_{1},\tau_{2},\ldots,\tau_{n})$. This gives a full classification of free, non-free and minimal multivariate skew polynomial rings over finite fields.

3 Explicit descriptions for finite fields

In the univariate case, $n=1$, explicit descriptions of all possible ring morphisms $\sigma:\mathbb{F}_{q}\longrightarrow\mathbb{F}_{q}$ and $\sigma$-derivations $\delta:\mathbb{F}_{q}\longrightarrow\mathbb{F}_{q}$ are well-known for any value of $q$.

First, all ring morphisms $\sigma:\mathbb{F}_{q}\longrightarrow\mathbb{F}_{q}$ form the Galois group of the field extension $\mathbb{F}_{p}\subseteq\mathbb{F}_{q}$. More concretely, they are given as follows (see [13, Th. 2.21]).

Proposition 1**.**

A map $\sigma:\mathbb{F}_{q}\longrightarrow\mathbb{F}_{q}$ is a ring morphism if, and only if, there exists an integer $0\leq j\leq m-1$ such that $\sigma(a)=a^{p^{j}}$, for all $a\in\mathbb{F}_{q}$, where $q=p^{m}$. In particular, $\sigma$ is a field automorphism.

Second, all $\sigma$-derivations over finite fields are inner derivations. More concretely, they are given as follows (see [5, Sec. 8.3] or [15, Sec. 4.2]).

Proposition 2**.**

Given a field automorphism $\sigma:\mathbb{F}_{q}\longrightarrow\mathbb{F}_{q}$, a map $\delta:\mathbb{F}_{q}\longrightarrow\mathbb{F}_{q}$ is a $\sigma$-derivation if, and only if, there exists an element $\lambda\in\mathbb{F}_{q}$ such that

[TABLE]

for all $a\in\mathbb{F}_{q}$. In particular, $\delta=0$ is the only ${\rm Id}$-derivation.

In the next two subsections, we will extend Propositions 1 and 2 to the case $n>1$.

3.1 All derivations are inner derivations

For technical reasons, we will first give descriptions of derivations, since they will prove to be useful to describe morphisms in the next subsection. In this subsection, we will show in Proposition 2 that all derivations are also inner derivations when $n\geq 1$. Furthermore, we will consider a more general definition of derivations and inner derivations that will also be useful in the next subsection.

Definition 1**.**

Given a field $\mathbb{F}$ and ring morphisms $\sigma,\tau:\mathbb{F}\longrightarrow\mathbb{F}^{n\times n}$, we say that a map $\delta:\mathbb{F}\longrightarrow\mathbb{F}^{n}$ is a $(\sigma,\tau)$-derivation if it is additive and satisfies that

[TABLE]

for all $a,b\in\mathbb{F}$. Moreover, we say that $\delta$ is an inner $(\sigma,\tau)$-derivation if there exists $\boldsymbol{\lambda}\in\mathbb{F}^{n}$ such that

[TABLE]

for all $a\in\mathbb{F}$.

Observe that a $\sigma$-derivation is just a $(\sigma,{\rm Id})$-derivation since the definition is given over a (commutative) field $\mathbb{F}$ and hence (2) holds. This is however not true over general division rings.

In addition, if $\mathbb{F}$ is a field and $\sigma(a)\tau(b)=\tau(b)\sigma(a)$, for all $a,b\in\mathbb{F}$ (which is always the case if $\tau={\rm Id}$ and $\mathbb{F}$ is a field), then an inner $(\sigma,\tau)$-derivation is indeed a $(\sigma,\tau)$-derivation. Again, the concept of inner $(\sigma,{\rm Id})$-derivation recovers the concept of inner $\sigma$-derivation if $\mathbb{F}$ is a field, but this is not the case if $\mathbb{F}$ is not commutative (see Definition 4 in Subsection 5.1 for the definition of inner derivations over general division rings).

The main result of this subsection is the following.

Theorem 1**.**

Let $\sigma,\tau:\mathbb{F}_{q}\longrightarrow\mathbb{F}_{q}^{n\times n}$ be ring morphisms. The following hold:

$\sigma(a)\tau(b)=\tau(b)\sigma(a)$* for all $a,b\in\mathbb{F}_{q}$ if, and only if, $\sigma(c)\tau(c)=\tau(c)\sigma(c)$ for a primitive element $c\in\mathbb{F}_{q}^{*}$.* 2. 2.

If Item 1 holds, then a map $\delta:\mathbb{F}_{q}\longrightarrow\mathbb{F}_{q}^{n}$ is a $(\sigma,\tau)$-derivation if, and only if, it is an inner $(\sigma,\tau)$-derivation.

In particular, for any ring morphism $\sigma:\mathbb{F}_{q}\longrightarrow\mathbb{F}_{q}^{n\times n}$, the only $(\sigma,\sigma)$-derivation is the zero derivation $\delta=0$.

The first item in Theorem 1 is straightforward to prove. We devote the rest of the section to proving the second item.

The following auxiliary lemma is obtained by applying (5) recursively on powers.

Lemma 1**.**

Given a field $\mathbb{F}$, ring morphisms $\sigma,\tau:\mathbb{F}\longrightarrow\mathbb{F}^{n\times n}$ such that $\sigma(a)\tau(a)=\tau(a)\sigma(a)$ for all $a\in\mathbb{F}$, and a $(\sigma,\tau)$-derivation $\delta:\mathbb{F}\longrightarrow\mathbb{F}^{n}$, it holds that

[TABLE]

for all $a\in\mathbb{F}$ and all integers $j\geq 0$.

Our second auxiliary lemma is the following fact about binomial coefficients over $\mathbb{F}_{q}$. Since we were not able to find an explicit proof in the literature, we provide one for convenience.

Lemma 2**.**

For every integer $0\leq i\leq q-1$, it holds that

[TABLE]

inside the finite field $\mathbb{F}_{q}$.

Proof.

In the conventional polynomial ring $\mathbb{F}_{q}[x,y]$ with commutative variables $xy=yx$, it holds that

[TABLE]

Cancelling $x-y\neq 0$ on both sides and applying Newton’s binomial formula for $(x-y)^{q-1}$, we conclude that

[TABLE]

and the result follows. ∎

With these tools we may now complete the proof of Theorem 1.

Proof of Theorem 1.

Let $c\in\mathbb{F}_{q}^{*}$ be a primitive element and define $S=\sigma(c)\in\mathbb{F}_{q}^{n\times n}$ and $T=\tau(c)\in\mathbb{F}_{q}^{n\times n}$. By Newton’s binomial formula and Lemma 2, it holds that

[TABLE]

Now, by Equation (8) and Lemma 1, we have that

[TABLE]

Define now the vector (note that $q\geq 2$)

[TABLE]

By definition of $\boldsymbol{\lambda}$, $S$ and $T$, and by (9), we have that

[TABLE]

Define then the map $\partial:\mathbb{F}_{q}\longrightarrow\mathbb{F}_{q}^{n}$ by

[TABLE]

for all $a\in\mathbb{F}_{q}$. Since $\sigma(a)\tau(b)=\tau(b)\sigma(a)$ for all $a,b\in\mathbb{F}_{q}$, by the first item in Theorem 1, we have that $\partial$ is a $(\sigma,\tau)$-derivation.

Finally, take any $a\in\mathbb{F}_{q}^{*}$. There exists an integer $0\leq j\leq q-1$ such that $a=c^{j+1}$. By Lemma 1 and the fact that $\delta(c)=\partial(c)$, we conclude that

[TABLE]

and we are done. ∎

Remark 1**.**

Observe that (10) gives an explicit formula for $\boldsymbol{\lambda}$ based on the evaluations of $\sigma$, $\tau$ and $\delta$ at a primitive element of $\mathbb{F}_{q}$.

Note that over infinite fields there exist non-zero ${\rm Id}$-derivations, which then cannot be inner, since any inner ${\rm Id}$-derivation over a field is necessarily zero. This holds for any $n\geq 1$.

Recall also that, if $n=1$, then all $\sigma$-derivations are inner $\sigma$-derivations even over infinite fields as long as $\sigma\neq{\rm Id}$ (see [5, Sec. 8.3] or [15, Prop. 39]). Interestingly, if $n>1$, there are $\sigma$-derivations that are not inner $\sigma$-derivations over infinite fields even if $\sigma\neq{\rm Id}$, as the next example shows.

Example 1**.**

Let $\mathbb{F}=\mathbb{F}_{p}(z)$ be the field of rational functions over $\mathbb{F}_{p}$, for a prime number $p$. Consider the ring morphism $\sigma:\mathbb{F}_{p}(z)\longrightarrow\mathbb{F}_{p}(z)^{2\times 2}$ given by

[TABLE]

for $f(z)\in\mathbb{F}_{p}(z)$, where $f(z)\mapsto f^{\prime}(z)$ is the usual derivation in $\mathbb{F}_{p}(z)$. Then the map $\delta:\mathbb{F}_{p}(z)\longrightarrow\mathbb{F}_{p}(z)^{2}$, given by

[TABLE]

is a $\sigma$-derivation that is not an inner $\sigma$-derivation. The proof is left to the reader.

3.2 All morphisms are diagonalizable

In this subsection, we will prove that we may “diagonalize” all ring morphisms $\sigma:\mathbb{F}_{q}\longrightarrow\mathbb{F}_{q}^{n\times n}$. The formal definition is as follows.

Definition 2**.**

Given an arbitrary division ring $\mathbb{F}$ and a ring morphism $\sigma:\mathbb{F}\longrightarrow\mathbb{F}^{n\times n}$, we say that $\sigma$ is diagonalizable if there exists an invertible matrix $A\in\mathbb{F}^{n\times n}$ and ring endomorphisms $\sigma_{1},\sigma_{2},\ldots,\sigma_{n}:\mathbb{F}\longrightarrow\mathbb{F}$ such that

[TABLE]

for all $a\in\mathbb{F}$.

Note that if $\tau:\mathbb{F}\longrightarrow\mathbb{F}^{n\times n}$ is a ring morphism over a division ring $\mathbb{F}$, then so is $\sigma:\mathbb{F}\longrightarrow\mathbb{F}^{n\times n}$ given by $\sigma(a)=A\tau(a)A^{-1}$, for all $a\in\mathbb{F}$, and for an invertible matrix $A\in\mathbb{F}^{n\times n}$. See also Proposition 3 in Subsection 4.1.

The main result of this section is the following.

Theorem 2**.**

Any ring morphism $\sigma:\mathbb{F}_{q}\longrightarrow\mathbb{F}_{q}^{n\times n}$ over a finite field $\mathbb{F}_{q}$ is diagonalizable.

Proof.

Fix a primitive element $c\in\mathbb{F}_{q}^{*}$, and define $S=\sigma(c)\in\mathbb{F}_{q}^{n\times n}$ and $F(t)=\det(S-tI)\in\mathbb{F}_{q}[t]$. Since any finite algebraic extension of a finite field is again a finite field, we have that $F(t)=\prod_{i=1}^{k}(t-\lambda_{i})^{n_{i}}$, where $\lambda_{1},\lambda_{2},\ldots,\lambda_{k}\in\mathbb{F}_{q^{r}}$ are pair-wise distinct, for a suitable finite field extension $\mathbb{F}_{q^{r}}$ of $\mathbb{F}_{q}$ and for positive integers $n_{1},n_{2},\ldots,n_{k}$ such that $n=n_{1}+n_{2}+\cdots+n_{k}$.

Hence $S$ admits a Jordan canonical form in $\mathbb{F}_{q^{r}}^{n\times n}$ (see for instance [4, Th. 25, p. 354] or [11, Sec. V.9]). In other words, there exists an invertible matrix $A\in\mathbb{F}_{q^{r}}^{n\times n}$ such that

[TABLE]

where the matrix $\Lambda_{i}$ is given by

[TABLE]

where $\varepsilon_{i,j}$ is either $1$ or [math], for $j=1,2,\ldots,n_{i}-1$, and for $i=1,2,\ldots,k$.

Now, for an arbitrary $a\in\mathbb{F}_{q}^{*}$, there exists an integer $0\leq j\leq q-1$ such that $a=c^{j}$. Using the fact that $\sigma(a)=\sigma(c)^{j}$ and the expressions (11) and (12), we see that there exist maps $\tau_{i}:\mathbb{F}_{q}\longrightarrow\mathbb{F}_{q^{r}}^{n_{i}\times n_{i}}$, for $i=1,2,\ldots,k$, such that

[TABLE]

for all $a\in\mathbb{F}_{q}$. Moreover, there exist functions $\varphi^{(i)}:\mathbb{F}_{q}\longrightarrow\mathbb{F}_{q^{r}}$ and $\varphi_{u,v}^{(i)}:\mathbb{F}_{q}\longrightarrow\mathbb{F}_{q^{r}}$, for $1\leq u<v\leq n_{i}$, such that

[TABLE]

for all $a\in\mathbb{F}_{q}$, and for $i=1,2,\ldots,k$.

Note that $\tau:\mathbb{F}_{q}\longrightarrow\mathbb{F}_{q^{r}}^{n\times n}$, given by $\tau(a)=A^{-1}\sigma(a)A$, for $a\in\mathbb{F}_{q}$, is a ring morphism, as observed in Proposition 3 in Subsection 4.1. In particular, the functions $\varphi^{(i)}:\mathbb{F}_{q}\longrightarrow\mathbb{F}_{q^{r}}$ are field morphisms, for $i=1,2,\ldots,k$. Thus $\varphi^{(i)}(a)^{q}=\varphi^{(i)}(a^{q})=\varphi^{(i)}(a)$ for all $a\in\mathbb{F}_{q}$, hence $\varphi^{(i)}:\mathbb{F}_{q}\longrightarrow\mathbb{F}_{q}$, and we deduce that $\lambda_{i}=\varphi^{(i)}(c)\in\mathbb{F}_{q}$ and, therefore, $\varphi_{u,v}^{(i)}:\mathbb{F}_{q}\longrightarrow\mathbb{F}_{q}$, for all $1\leq u<v\leq n_{i}$ and for all $i=1,2,\ldots,k$. In other words, $\tau_{i}:\mathbb{F}_{q}\longrightarrow\mathbb{F}_{q}^{n_{i}\times n_{i}}$, for all $i=1,2,\ldots,k$, and $\tau:\mathbb{F}_{q}\longrightarrow\mathbb{F}_{q}^{n\times n}$. In particular, $\sigma(c),\tau(c)\in\mathbb{F}_{q}^{n\times n}$ are similar over $\mathbb{F}_{q^{r}}$ (i.e., $\sigma(c)=A\tau(c)A^{-1}$ and $A\in\mathbb{F}_{q^{r}}^{n\times n}$ is invertible), hence they are also similar over $\mathbb{F}_{q}$ (since they share the same rational canonical form, see [4, Th. 42, p. 353]), and therefore we may assume that $A\in\mathbb{F}_{q}^{n\times n}$.

Next, the fact that $\tau_{i}:\mathbb{F}_{q}\longrightarrow\mathbb{F}_{q}^{n_{i}\times n_{i}}$ is a ring morphism implies that $\varphi_{j,j+1}^{(i)}:\mathbb{F}_{q}\longrightarrow\mathbb{F}_{q}$ are $(\varphi^{(i)},\varphi^{(i)})$-derivations, for $j=1,2,\ldots,n_{i}-1$, and for $i=1,2,\ldots,k$. By Theorem 1, it follows that $\varphi_{j,j+1}^{(i)}(a)=0$ for all $a\in\mathbb{F}_{q}$, and in particular, $\varepsilon_{i,j}=\varphi_{j,j+1}^{(i)}(c)=0$, for all $j=1,2,\ldots,n_{i}-1$ and all $i=1,2,\ldots,k$. Hence, we conclude that

[TABLE]

for all $a\in\mathbb{F}_{q}$, and the result follows. ∎

Remark 2**.**

Observe that the proof above gives an algorithmic method to diagonalize $\sigma$, since it is based on diagonalizing the matrix $\sigma(c)\in\mathbb{F}_{q}^{n\times n}$ for a primitive element $c\in\mathbb{F}_{q}^{*}$.

Remark 3**.**

Recall that, although any square matrix over a finite field admits a Jordan canonical form over its algebraic closure (thus over a finite extension), when it comes to diagionalization, any scenario may happen. As simple examples, the matrix

[TABLE]

is diagonalizable over $\mathbb{F}_{27}$ but not over $\mathbb{F}_{3}$, and the matrix

[TABLE]

is not diagonalizable over any field extension of $\mathbb{F}_{2}$, since it is non-zero and nilpotent.

Note that the well-known Skolem-Noether theorem [24] cannot be directly applied to prove Theorem 2. Following [17, Th. 2.10], the theorem in general form reads as follows.

Theorem 3** (Skolem-Noether).**

Let $\mathbb{F}$ be a field. Given simple (and possibly non-commutative) $\mathbb{F}$-algebras $\mathcal{A}$ and $\mathcal{B}$ such that $\mathbb{F}$ is the center of $\mathcal{B}$, then for any two $\mathbb{F}$-algebra morphisms $\sigma,\tau:\mathcal{A}\longrightarrow\mathcal{B}$, there exists an invertible element $A\in\mathcal{B}$ such that

[TABLE]

for all $a\in\mathcal{A}$.

Observe that the assumptions in Theorem 3 do not hold in the case of Theorem 2 when setting $\mathcal{A}=\mathbb{F}_{q}$ and $\mathcal{B}=\mathbb{F}_{q}^{n\times n}$ (resp. $\mathcal{B}=\sigma(\mathbb{F}_{q})\subseteq\mathbb{F}_{q}^{n\times n}$). First, the center of $\mathbb{F}_{q}^{n\times n}$ contains $\mathbb{F}_{q}$ (resp. $\sigma(\mathbb{F}_{q})$). However, the map $\sigma$ in Theorem 2 is not linear over $\mathbb{F}_{q}$ (resp. $\sigma(\mathbb{F}_{q})\cong\mathbb{F}_{q}$) in general. Hence the assumptions in Theorem 3 do not hold.

In fact, it is not true that any two ring morphisms $\sigma,\tau:\mathbb{F}_{q}\longrightarrow\mathbb{F}_{q}^{n\times n}$ are related as in (13), even if $\sigma(\mathbb{F}_{q})=A\tau(\mathbb{F}_{q})A^{-1}$ for all invertible matrices $A\in\mathbb{F}_{q}^{n\times n}$. We show this in the next example. Note that this observation is trivial when $n=1$.

Example 2**.**

We will fix $n=2$ for illustration purposes, although this example works for any $n\geq 1$ and it is trivial for $n=1$. Let $q=p^{4}$ for some prime number $p$, and let $\sigma:\mathbb{F}_{q}\longrightarrow\mathbb{F}_{q}^{2\times 2}$ be given by

[TABLE]

for all $a\in\mathbb{F}_{q}$. Since $(a^{p^{2}})^{p^{2}}=a^{p^{4}}=a$, for all $a\in\mathbb{F}_{q}$, we deduce that $\sigma(\mathbb{F}_{q})=\{aI\in\mathbb{F}_{q}^{2\times 2}\mid a\in\mathbb{F}_{q}\}$. Let $\tau={\rm Id}:\mathbb{F}_{q}\longrightarrow\mathbb{F}_{q}^{2\times 2}$. Obviously, it holds that $\sigma(\mathbb{F}_{q})=\tau(\mathbb{F}_{q})$, or even further, $\sigma(\mathbb{F}_{q})=A\tau(\mathbb{F}_{q})A^{-1}$, for all invertible matrices $A\in\mathbb{F}_{q}^{2\times 2}$. However, if there existed an invertible matrix $A\in\mathbb{F}_{q}^{2\times 2}$ such that $\sigma(a)=A\tau(a)A^{-1}$, for all $a\in\mathbb{F}_{q}$, then the elements fixed by $\sigma$ (i.e., those $a\in\mathbb{F}_{q}$ such that $\sigma(a)=aI$) would be the same as those fixed by $\tau$. This is not true, as the elements fixed by $\sigma$ form $\mathbb{F}_{p^{2}}$, whereas those fixed by $\tau$ form $\mathbb{F}_{p^{4}}$.

Example 2 also shows that we lose information on similarity when looking at the set $\sigma(\mathbb{F}_{q})\subseteq\mathbb{F}_{q}^{n\times n}$. Observe that, if $\mathbb{F}$ is a field and $\sigma:\mathbb{F}\longrightarrow\mathbb{F}^{n\times n}$ is a ring morphism, then $\sigma(\mathbb{F})$ is a subfield of the ring $\mathbb{F}^{n\times n}$. Subfields of $\mathbb{F}^{n\times n}$ are generally called matrix fields. Two subfields $\mathcal{A},\mathcal{B}\subseteq\mathbb{F}^{n\times n}$ are called similar if there exists an invertible matrix $A\in\mathbb{F}^{n\times n}$ such that $\mathcal{A}=A\mathcal{B}A^{-1}$. Matrix fields over finite fields and their similarity have been studied earlier, for instance in [2, 3]. However, as Example 2 shows, it may hold that $\sigma(\mathbb{F}_{q}),\tau(\mathbb{F}_{q})\subseteq\mathbb{F}_{q}^{n\times n}$ are similar matrix fields (or even equal), but still $\sigma,\tau:\mathbb{F}_{q}\longrightarrow\mathbb{F}_{q}^{n\times n}$ are not similar ring morphisms (i.e., there is no invertible matrix $A\in\mathbb{F}_{q}^{n\times n}$ such that $\sigma(a)=A\tau(a)A^{-1}$, for all $a\in\mathbb{F}_{q}$).

We conclude by showing that such a diagonalization of ring morphisms $\sigma:\mathbb{F}\longrightarrow\mathbb{F}^{n\times n}$ is not always possible over infinite fields.

Example 3**.**

Let $\mathbb{F}=\mathbb{F}_{p}(z)$ be the field of rational functions over $\mathbb{F}_{p}$, for a prime number $p$. The ring morphism $\sigma:\mathbb{F}_{p}(z)\longrightarrow\mathbb{F}_{p}(z)^{2\times 2}$ given by

[TABLE]

for $f(z)\in\mathbb{F}_{p}(z)$, as in Example 1, is not diagonalizable: Similar to the previous example, the subfield of $\mathbb{F}_{p}(z)$ fixed by $\sigma$ is $\mathbb{F}_{p}(z^{p})$, but there is no field endomorphism of $\mathbb{F}_{p}(z)$, other than the identity, leaving the elements in $\mathbb{F}_{p}(z^{p})$ fixed (see also [15, Ex. 48]).

4 Linear transformations of variables

In the previous section, we showed that all ring morphisms $\sigma:\mathbb{F}_{q}\longrightarrow\mathbb{F}_{q}^{n\times n}$ over a finite field $\mathbb{F}_{q}$ are diagonalizable (Theorem 2), that is, they are similar (i.e., as in Proposition 3 below) to a diagonal ring morphism.

In this section, we show that similar ring morphisms allow to naturally define linear transformations of variables between the corresponding multivariate skew polynomial rings. Furthermore, such transformations are ring isomorphisms and preserve evaluations and degrees, hence can be naturally extended to non-free multivariate skew polynomial rings (see Section 2). Combined with Theorem 2, these linear transformations will show in Section 7 how to simplify and classify multivariate skew polynomial rings over finite fields.

Each subsection is devoted to a different property of the mentioned ring isomorphism, which will be denoted by $\varphi_{A}$ and depends on the invertible matrix $A\in\mathbb{F}^{n\times n}$ that gives the linear transformation. Throughout this section, $\mathbb{F}$ may be any division ring.

4.1 Definition of the map $\varphi_{A}$

We start by formalizing the following observation, which we have mentioned earlier in the paper.

Proposition 3**.**

Let $\tau:\mathbb{F}\longrightarrow\mathbb{F}^{n\times n}$ be a ring morphism and let $\delta_{\tau}:\mathbb{F}\longrightarrow\mathbb{F}^{n}$ be a $\tau$-derivation. For an invertible matrix $A\in\mathbb{F}^{n\times n}$, define the maps $\sigma:\mathbb{F}\longrightarrow\mathbb{F}^{n\times n}$ and $\delta_{\sigma}:\mathbb{F}\longrightarrow\mathbb{F}^{n}$ by

[TABLE]

for all $a\in\mathbb{F}$. Then $\sigma$ a ring morphism and $\delta$ is a $\sigma$-derivation.

Linear transformations of variables are then defined as follows.

Definition 3**.**

Let $\sigma,\tau:\mathbb{F}\longrightarrow\mathbb{F}^{n\times n}$ be ring morphisms and let $\delta_{\sigma},\delta_{\tau}:\mathbb{F}\longrightarrow\mathbb{F}^{n}$ be a $\sigma$-derivation and a $\tau$-derivation, respectively, as in the previous proposition, for an invertible matrix $A\in\mathbb{F}^{n\times n}$. We define the map

[TABLE]

as follows. First, we define $\varphi_{A}(1)=1$ and

[TABLE]

Next we define $\varphi_{A}(\mathfrak{m}(\mathbf{x}))$ recursively on monomials $\mathfrak{m}(\mathbf{x})\in\mathcal{M}$. Assume that $\varphi_{A}(\mathfrak{m}(\mathbf{x}))$ is defined, for a given $\mathfrak{m}(\mathbf{x})\in\mathcal{M}$. Then we define

[TABLE]

Finally, if $F(\mathbf{x})=\sum_{\mathfrak{m}(\mathbf{x})\in\mathcal{M}}F_{\mathfrak{m}}\mathfrak{m}(\mathbf{x})\in\mathbb{F}[\mathbf{x};\sigma,\delta_{\sigma}]$, where $F_{\mathfrak{m}}\in\mathbb{F}$, for all $\mathfrak{m}(\mathbf{x})\in\mathcal{M}$, we define

[TABLE]

Given $F(\mathbf{x})\in\mathbb{F}[\mathbf{x};\sigma,\delta_{\sigma}]$ or $\mathfrak{m}(\mathbf{x})\in\mathcal{M}$, we will use the notation

[TABLE]

Remark 4**.**

Observe that, if $\mathbb{F}$ is a field and $\tau={\rm Id}$, then $\sigma=\tau$ and $\delta_{\sigma}=A\delta_{\tau}$ represents a linear transformation of the standard derivations in $\delta_{\tau}$. In case $\delta_{\tau}=0$, then $\delta_{\sigma}=0$, and both skew polynomial rings in (14) are the conventional free multivariate polynomial ring $\mathbb{F}[\mathbf{x}]$. In that case, $\varphi_{A}(F(\mathbf{x}))$ coincides with the usual definition of $F(A\mathbf{x})$ that consists in substituting $\mathbf{x}$ by $A\mathbf{x}$, for any $F(\mathbf{x})\in\mathbb{F}[\mathbf{x}]$. Observe that if $\mathbb{F}$ is a non-commutative division ring and $\tau={\rm Id}$, then the ring morphism $\sigma:\mathbb{F}\longrightarrow\mathbb{F}^{n\times n}$ given by $\sigma(a)=A(aI)A^{-1}$, for $a\in\mathbb{F}$, is not necessarily the identity if the entries of $A$ do not lie in the center of $\mathbb{F}$.

4.2 The map $\varphi_{A}$ is additive and multiplicative

In this subsection, we will prove that the map $\varphi_{A}$ in Definition 3 is a ring morphism.

Proposition 4**.**

Let $\sigma,\tau:\mathbb{F}\longrightarrow\mathbb{F}^{n\times n}$ be ring morphisms and let $\delta_{\sigma},\delta_{\tau}:\mathbb{F}\longrightarrow\mathbb{F}^{n}$ be a $\sigma$-derivation and a $\tau$-derivation, respectively, as in Proposition 3, for an invertible matrix $A\in\mathbb{F}^{n\times n}$. The map $\varphi_{A}:\mathbb{F}[\mathbf{x};\sigma,\delta_{\sigma}]\longrightarrow\mathbb{F}[\mathbf{x};\tau,\delta_{\tau}]$ in Definition 3 is a ring morphism. That is, with notation as in (16), it holds that

[TABLE]

for all $F(\mathbf{x}),G(\mathbf{x})\in\mathbb{F}[\mathbf{x};\sigma,\delta_{\sigma}]$, where the operations on the right-hand sides are in $\mathbb{F}[\mathbf{x};\tau,\delta_{\tau}]$, and the operations on the left-hand sides are in $\mathbb{F}[\mathbf{x};\sigma,\delta_{\sigma}]$.

Proof.

The fact that $\varphi_{A}$ is additive (or even left linear over $\mathbb{F}$) follows directly from the definitions.

Now, due to (15), in order to prove that $\varphi_{A}$ is multiplicative, we only need to show that

[TABLE]

for all $\mathfrak{m}(\mathbf{x})\in\mathcal{M}$ and all $G(\mathbf{x})\in\mathbb{F}[\mathbf{x};\sigma,\delta_{\sigma}]$. We do this recursively on $\mathfrak{m}(\mathbf{x})\in\mathcal{M}$.

First observe that it is trivial for $\mathfrak{m}(\mathbf{x})=1$. To perform the induction step, we need to prove the result for $\mathfrak{m}(\mathbf{x})=x_{i}$, for $i=1,2,\ldots,n$. Denote $G(\mathbf{x})=\sum_{\mathfrak{m}(\mathbf{x})\in\mathcal{M}}G_{\mathfrak{m}}\mathfrak{m}(\mathbf{x})$, where $G_{\mathfrak{m}}\in\mathbb{F}$, for all $\mathfrak{m}(\mathbf{x})\in\mathcal{M}$. First it holds that

[TABLE]

since these operations are in the ring $\mathbb{F}[\mathbf{x};\sigma,\delta_{\sigma}]$. Therefore, by definition, it holds that

[TABLE]

Next, working in the ring $\mathbb{F}[\mathbf{x};\tau,\delta_{\tau}]$, we see that

[TABLE]

[TABLE]

[TABLE]

Thus we deduce that

[TABLE]

where operations on the right-hand side are in the ring $\mathbb{F}[\mathbf{x};\tau,\delta_{\tau}]$.

Finally, assume that the result holds for a given $\mathfrak{m}(\mathbf{x})\in\mathcal{M}$, and we will prove it for $x_{i}\mathfrak{m}(\mathbf{x})$, for $i=1,2,\ldots,n$. It follows from (17) and the hypothesis on $\mathfrak{m}(\mathbf{x})$ that

[TABLE]

[TABLE]

and we are done. ∎

4.3 The map $\varphi_{A}$ preserves evaluations

In this subsection, we show that the $(\tau,\delta_{\tau})$-evaluation of $F(A\mathbf{x})\in\mathbb{F}[\mathbf{x};\tau,\delta_{\tau}]$ at an affine point $\mathbf{a}\in\mathbb{F}^{n}$ coincides with the $(\sigma,\delta_{\sigma})$-evaluation of $F(\mathbf{x})\in\mathbb{F}[\mathbf{x};\sigma,\delta_{\sigma}]$ at the affine point $A\mathbf{a}\in\mathbb{F}^{n}$. See (3) in Section 2 for the definition of evaluation.

Proposition 5**.**

Let $\sigma,\tau:\mathbb{F}\longrightarrow\mathbb{F}^{n\times n}$ be ring morphisms and let $\delta_{\sigma},\delta_{\tau}:\mathbb{F}\longrightarrow\mathbb{F}^{n}$ be a $\sigma$-derivation and a $\tau$-derivation, respectively, as in Proposition 3, for an invertible matrix $A\in\mathbb{F}^{n\times n}$. The map $\varphi_{A}:\mathbb{F}[\mathbf{x};\sigma,\delta_{\sigma}]\longrightarrow\mathbb{F}[\mathbf{x};\tau,\delta_{\tau}]$ in Definition 3 preserves evaluations over any point $\mathbf{a}\in\mathbb{F}^{n}$ after multiplication with the matrix $A$. More concretely, according to the definition of evaluation in (3), it holds that

[TABLE]

for all $F(\mathbf{x})\in\mathbb{F}[\mathbf{x};\sigma,\delta_{\sigma}]$ and all $\mathbf{a}\in\mathbb{F}^{n}$. In other words, the evaluation of $F(A\mathbf{x})$ in the point $\mathbf{a}$ is $F(A\mathbf{a})$.

Proof.

Fix $F(\mathbf{x})\in\mathbb{F}[\mathbf{x};\sigma,\delta_{\sigma}]$ and $\mathbf{a}\in\mathbb{F}^{n}$, and denote $\mathbf{b}=(b_{1},b_{2},\ldots,b_{n})=A\mathbf{a}\in\mathbb{F}^{n}$. By the results in Section 2, there exist $G_{1}(\mathbf{x}),G_{2}(\mathbf{x}),\ldots,$ $G_{n}(\mathbf{x})\in\mathbb{F}[\mathbf{x};\sigma,\delta_{\sigma}]$ such that

[TABLE]

where $\mathbf{G}(\mathbf{x})\in\mathbb{F}[\mathbf{x};\sigma,\delta_{\sigma}]^{n}$ is the column vector of length $n$ whose $i$th row is $G_{i}(\mathbf{x})$, for $i=1,2,\ldots,n$. Now, using that $\varphi_{A}$ is a ring morphism (Proposition 4), we deduce that

[TABLE]

and the result follows by the uniqueness of the remainder $F(A\mathbf{a})\in\mathbb{F}$ (see Section 2). ∎

Thanks to this result, we may extend $\varphi_{A}$ to non-free multivariate skew polynomial rings. We follow the definitions in Section 2. However, for clarity, we denote by $I^{\tau,\delta_{\tau}}(\mathbb{F}^{n})$ and $I^{\sigma,\delta_{\sigma}}(\mathbb{F}^{n})$ the two-sided ideals of $\mathbb{F}[\mathbf{x};\tau,\delta_{\tau}]$ and $\mathbb{F}[\mathbf{x};\sigma,\delta_{\sigma}]$, respectively, given by skew polynomials that vanish at every point. We have the following consequence of Proposition 5.

Corollary 1**.**

Let $\sigma,\tau:\mathbb{F}\longrightarrow\mathbb{F}^{n\times n}$ be ring morphisms and let $\delta_{\sigma},\delta_{\tau}:\mathbb{F}\longrightarrow\mathbb{F}^{n}$ be a $\sigma$-derivation and a $\tau$-derivation, respectively, as in Proposition 3, for an invertible matrix $A\in\mathbb{F}^{n\times n}$. It holds that

[TABLE]

Therefore, we may extend $\varphi_{A}$ to a ring morphism between non-free multivariate skew polynomial rings, that is, to ring morphisms

[TABLE]

for any two-sided ideal $I\subseteq I^{\sigma,\delta_{\sigma}}(\mathbb{F}^{n})$, where $J=\varphi_{A}(I)\subseteq I^{\tau,\delta_{\tau}}(\mathbb{F}^{n})$ is also a two-sided ideal. Furthermore, evaluations are also preserved by $\varphi_{A}$ as in (18) over non-free multivariate skew polynomial rings.

We conclude with a remark on conjugacy and the so-called product rule. Two affine points $\mathbf{a},\mathbf{b}\in\mathbb{F}^{n}$ are said to be $(\tau,\delta_{\tau})$-conjugate if there exists $c\in\mathbb{F}^{*}$ such that

[TABLE]

This concept was introduced in [9, Eq. (2.5)] when $n=1$, and in [16, Def. 11] in the general case. With notation as in Proposition 3, it is easy to check that $\mathbf{a}$ and $\mathbf{b}$ are $(\tau,\delta_{\tau})$-conjugate if, and only if, $A\mathbf{a}$ and $A\mathbf{b}$ are $(\sigma,\delta_{\sigma})$-conjugate, with the same element $c\in\mathbb{F}^{*}$ in the conjugacy relation (20) in both cases.

The conjugacy relation allows to relate evaluations and products through the product rule, which was given in [9, Th. 2.7] when $n=1$, and in [16, Th. 3] in the general case. This result is as follows. Let $F(\mathbf{x}),G(\mathbf{x})\in\mathbb{F}[\mathbf{x};\tau,\delta_{\tau}]$, let $\mathbf{a}\in\mathbb{F}^{n}$, and let $c=G(\mathbf{a})$. If $c=0$, then $(FG)(\mathbf{a})=E^{\tau,\delta_{\tau}}_{\mathbf{a}}((FG)(\mathbf{x}))=0$, and if $c\neq 0$, then

[TABLE]

where $\mathbf{a}$ and $\mathbf{b}$ are $(\tau,\delta_{\tau})$-conjugate with $c$ in the conjugacy relation (20).

Therefore, Propositions 4 and 5 are consistent with the product rule, since the following identities, which need to hold, actually hold:

[TABLE]

[TABLE]

assuming that $c=E^{\sigma,\delta_{\sigma}}_{A\mathbf{a}}(G(\mathbf{x}))=E^{\tau,\delta_{\tau}}_{\mathbf{a}}(G(A\mathbf{x}))\neq 0$, where $\mathbf{a}$ and $\mathbf{b}$ are $(\tau,\delta_{\tau})$-conjugate, thus $A\mathbf{a}$ and $A\mathbf{b}$ are $(\sigma,\delta_{\sigma})$-conjugate, both using $c$ in the conjugacy relation. Similarly if $c=0$.

4.4 The inverse of the map $\varphi_{A}$

In this subsection, we prove that the ring morphism $\varphi_{A}$ has $\varphi_{A^{-1}}$ as its inverse, and thus it is a ring isomorphism. The main result is the following.

Proposition 6**.**

Let $\sigma,\tau,\upsilon:\mathbb{F}\longrightarrow\mathbb{F}^{n\times n}$ be ring morphisms and let $\delta_{\sigma},\delta_{\tau},\delta_{\upsilon}:\mathbb{F}\longrightarrow\mathbb{F}^{n}$ be a $\sigma$-derivation, a $\tau$-derivation and a $\upsilon$-derivation, respectively, where

[TABLE]

[TABLE]

for all $a\in\mathbb{F}$, for invertible matrices $A,B\in\mathbb{F}^{n\times n}$. It holds that

[TABLE]

In other words, for any $F(\mathbf{x})\in\mathbb{F}[\mathbf{x};\sigma,\delta_{\sigma}]$ and using notation as in (16), it holds that

[TABLE]

in the ring $\mathbb{F}[\mathbf{x};\upsilon,\delta_{\upsilon}]$.

Proof.

By the linearity of the maps $\varphi_{A}$ and $\varphi_{B}$, we only need to prove this result for $F(\mathbf{x})=\mathfrak{m}(\mathbf{x})$, for $\mathfrak{m}(\mathbf{x})\in\mathcal{M}$. We do this recursively on $\mathfrak{m}(\mathbf{x})\in\mathcal{M}$.

First, the result is trivial for $\mathfrak{m}(\mathbf{x})=1$. Next, we have that

[TABLE]

thus the result holds for $\mathfrak{m}(\mathbf{x})=x_{i}$, for $i=1,2,\ldots,n$. Finally, assume that the result holds for a given $\mathfrak{m}(\mathbf{x})\in\mathcal{M}$, and we will prove it for $x_{i}\mathfrak{m}(\mathbf{x})$, for $i=1,2,\ldots,n$. Let $\mathbf{F}(\mathbf{x})=\mathbf{x}\mathfrak{m}(\mathbf{x})\in\mathbb{F}[\mathbf{x};\sigma,\delta_{\sigma}]^{n}$ and denote $\mathbf{G}(\mathbf{x})=\mathbf{F}(A\mathbf{x})\in\mathbb{F}[\mathbf{x};\tau,\delta_{\tau}]^{n}$. By Definition 3, $\mathbf{G}(\mathbf{x})=(A\mathbf{x})\mathfrak{m}(A\mathbf{x})$. By (21) and the hypothesis on $\mathfrak{m}(\mathbf{x})$, we conclude that

[TABLE]

[TABLE]

and we are done. ∎

Moreover, we have the following obvious fact.

Proposition 7**.**

Given a ring morphism $\sigma:\mathbb{F}\longrightarrow\mathbb{F}^{n\times n}$ and a $\sigma$-derivation $\delta:\mathbb{F}\longrightarrow\mathbb{F}^{n}$, it holds that

[TABLE]

is the identity morphism, where $I\in\mathbb{F}^{n\times n}$ is the identity matrix.

Therefore, we conclude that $\varphi_{A}$ is a ring isomorphism with inverse given by $\varphi_{A^{-1}}$.

Corollary 2**.**

Let $\sigma,\tau:\mathbb{F}\longrightarrow\mathbb{F}^{n\times n}$ be ring morphisms and let $\delta_{\sigma},\delta_{\tau}:\mathbb{F}\longrightarrow\mathbb{F}^{n}$ be a $\sigma$-derivation and a $\tau$-derivation, respectively, as in Proposition 3, for an invertible matrix $A\in\mathbb{F}^{n\times n}$. Observe that

[TABLE]

for all $a\in\mathbb{F}$. We conclude that

[TABLE]

That is, $\varphi_{A}^{-1}=\varphi_{A^{-1}}$, and $\varphi_{A}$ is a ring isomorphism.

Remark 5**.**

Observe that the results in this section hold automatically over non-free multivariate skew polynomial rings due to Corollary 1.

4.5 The map $\varphi_{A}$ preserves degrees

We conclude by showing that $\varphi_{A}$ preserves degrees.

Proposition 8**.**

Let $\sigma,\tau:\mathbb{F}\longrightarrow\mathbb{F}^{n\times n}$ be ring morphisms and let $\delta_{\sigma},\delta_{\tau}:\mathbb{F}\longrightarrow\mathbb{F}^{n}$ be a $\sigma$-derivation and a $\tau$-derivation, respectively, as in Proposition 3, for an invertible matrix $A\in\mathbb{F}^{n\times n}$. It holds that

[TABLE]

for all $F(\mathbf{x})\in\mathbb{F}[\mathbf{x};\sigma,\delta_{\sigma}]$.

Proof.

From Definition 3, it is easy to check that $\deg(\varphi_{A}(F(\mathbf{x})))\leq\deg(F(\mathbf{x}))$, for all $F(\mathbf{x})\in\mathbb{F}[\mathbf{x};\sigma,\delta_{\sigma}]$. Thus, it follows from Corollary 2 that

[TABLE]

and the result follows. ∎

5 Translations of variables

In Subsection 3.1, we showed that all derivations are inner derivations over finite fields (Theorem 1).

In this section, we study translations of variables over multivariate skew polynomial rings with the same ring morphism $\sigma:\mathbb{F}\longrightarrow\mathbb{F}^{n\times n}$ and whose $\sigma$-derivations differ additively by an inner $\sigma$-derivation. Such transformations of variables yield again ring isomorphisms between multivariate skew polynomial rings. By composing them with the linear transformations from the previous section, we will obtain in Section 6 a complete collection of affine transformations of variables between the corresponding multivariate skew polynomial rings. Combined with Theorem 1, these translations of variables will show in Section 6 how to simplify multivariate skew polynomial rings over finite fields.

Again, translations of variables preserve evaluations and degrees of free multivariate skew polynomials, hence they can be extended to non-free multivariate skew polynomial rings. We will also divide the section in different subsections, each devoted to a different property of the ring isomorphism, which we now denote by $\phi_{\boldsymbol{\lambda}}$ and which depends on the translation vector $\boldsymbol{\lambda}\in\mathbb{F}^{n}$.

5.1 Definition of the map $\phi_{\boldsymbol{\lambda}}$

As mentioned in Subsection 3.1, the definition of inner derivations in Definition 1 gives derivations only over fields. A slight modification, first given in [16, Example 2] when $n>1$, allows to consider inner derivations over any division ring.

Definition 4**.**

Given a ring morphism $\sigma:\mathbb{F}\longrightarrow\mathbb{F}^{n\times n}$, we say that $\delta:\mathbb{F}\longrightarrow\mathbb{F}^{n}$ is an inner $\sigma$-derivation if there exists $\boldsymbol{\lambda}\in\mathbb{F}^{n}$ such that

[TABLE]

for all $a\in\mathbb{F}$.

Observe that, with this definition, an inner $\sigma$-derivation is indeed a $\sigma$-derivation, since it satisfies (2) even if $\mathbb{F}$ is non-commutative.

Translations of variables are then defined as follows.

Definition 5**.**

Fix a ring morphism $\sigma:\mathbb{F}\longrightarrow\mathbb{F}^{n\times n}$ and $\sigma$-derivations $\delta,\delta^{\prime}:\mathbb{F}\longrightarrow\mathbb{F}^{n}$ such that $\delta-\delta^{\prime}$ is an inner $\sigma$-derivation, that is,

[TABLE]

for $a\in\mathbb{F}$, for a given $\boldsymbol{\lambda}\in\mathbb{F}^{n}$. We then define the map

[TABLE]

as follows. First, we define $\phi_{\boldsymbol{\lambda}}(1)=1$ and

[TABLE]

Next we define $\phi_{\boldsymbol{\lambda}}(\mathfrak{m}(\mathbf{x}))$ recursively on monomials $\mathfrak{m}(\mathbf{x})\in\mathcal{M}$. Assume that $\phi_{\boldsymbol{\lambda}}(\mathfrak{m}(\mathbf{x}))$ is defined, for a given $\mathfrak{m}(\mathbf{x})\in\mathcal{M}$. Then define

[TABLE]

Finally, if $F(\mathbf{x})=\sum_{\mathfrak{m}(\mathbf{x})\in\mathcal{M}}F_{\mathfrak{m}}\mathfrak{m}(\mathbf{x})\in\mathbb{F}[\mathbf{x};\sigma,\delta]$, where $F_{\mathfrak{m}}\in\mathbb{F}$, for all $\mathfrak{m}(\mathbf{x})\in\mathcal{M}$, then we define

[TABLE]

Given $F(\mathbf{x})\in\mathbb{F}[\mathbf{x};\sigma,\delta]$ or $\mathfrak{m}(\mathbf{x})\in\mathcal{M}$, we will use the notation

[TABLE]

Remark 6**.**

Note that, if $\mathbb{F}$ is a field and $\sigma={\rm Id}$, then $\delta=\delta^{\prime}$. In particular, if $\delta=\delta^{\prime}=0$, then both multivariate skew polynomial rings in (22) are the conventional free multivariate polynomial ring $\mathbb{F}[\mathbf{x}]$. Moreover, in that case $\phi_{\boldsymbol{\lambda}}$ coincides with the usual definition of $F(\mathbf{x}+\boldsymbol{\lambda})$ that consists in substituting $\mathbf{x}$ by $\mathbf{x}+\boldsymbol{\lambda}$, for any $F(\mathbf{x})\in\mathbb{F}[\mathbf{x}]$. Observe that if $\mathbb{F}$ is non-commutative and $\boldsymbol{\lambda}\neq\mathbf{0}$, then $\delta\neq\delta^{\prime}$ since the map $a\mapsto\boldsymbol{\lambda}a-a\boldsymbol{\lambda}$, for $a\in\mathbb{F}$, is not necessarily zero if the components of $\boldsymbol{\lambda}$ do not lie in the center of $\mathbb{F}$.

5.2 The map $\phi_{\boldsymbol{\lambda}}$ is additive and multiplicative

In this subsection, we will prove that the map $\phi_{\boldsymbol{\lambda}}$ in Definition 5 is a ring morphism. The proofs of the properties of $\phi_{\boldsymbol{\lambda}}$ are similar to those of $\varphi_{A}$ in Section 4. However, for clarity, we include the proof of the following result.

Proposition 9**.**

Let $\sigma:\mathbb{F}\longrightarrow\mathbb{F}^{n\times n}$ be a ring morphism and let $\delta,\delta^{\prime}:\mathbb{F}\longrightarrow\mathbb{F}^{n}$ be $\sigma$-derivations as in Definition 5, for a given $\boldsymbol{\lambda}\in\mathbb{F}^{n}$. Then the map $\phi_{\boldsymbol{\lambda}}:\mathbb{F}[\mathbf{x};\sigma,\delta]\longrightarrow\mathbb{F}[\mathbf{x};\sigma,\delta^{\prime}]$ in Definition 5 is a ring morphism. That is, with notation as in (24), it holds that

[TABLE]

for all $F(\mathbf{x}),G(\mathbf{x})\in\mathbb{F}[\mathbf{x};\sigma,\delta]$, where operations on the right-hand sides are in the ring $\mathbb{F}[\mathbf{x};\sigma,\delta^{\prime}]$, and operations on the left-hand sides are in the ring $\mathbb{F}[\mathbf{x};\sigma,\delta]$.

Proof.

The fact that $\phi_{\boldsymbol{\lambda}}$ is additive (or even left linear over $\mathbb{F}$) follows directly from the definitions.

Because of (23), to prove that $\phi_{\boldsymbol{\lambda}}$ is multiplicative, we only need to show that

[TABLE]

for all $\mathfrak{m}(\mathbf{x})\in\mathcal{M}$ and all $G(\mathbf{x})\in\mathbb{F}[\mathbf{x};\sigma,\delta]$. We do this recursively on $\mathfrak{m}(\mathbf{x})\in\mathcal{M}$.

First, the result is trivial for $\mathfrak{m}(\mathbf{x})=1$. Following the proof of Proposition 4, we need to prove the result for $\mathfrak{m}(\mathbf{x})=x_{i}$, for all $i=1,2,\ldots,n$, to perform the induction step. Denote $G(\mathbf{x})=\sum_{\mathfrak{m}(\mathbf{x})\in\mathcal{M}}G_{\mathfrak{m}}\mathfrak{m}(\mathbf{x})$, where $G_{\mathfrak{m}}\in\mathbb{F}$, for all $\mathfrak{m}(\mathbf{x})\in\mathcal{M}$. First, it holds that

[TABLE]

since these operations are in the ring $\mathbb{F}[\mathbf{x};\sigma,\delta]$. Thus by definition, it holds that

[TABLE]

Next, working in the ring $\mathbb{F}[\mathbf{x};\sigma,\delta^{\prime}]$, we have that

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Therefore, we deduce that

[TABLE]

where operations on the right-hand side are in the ring $\mathbb{F}[\mathbf{x};\sigma,\delta^{\prime}]$.

Finally, assume that the result holds for a given $\mathfrak{m}(\mathbf{x})\in\mathcal{M}$. To prove it for $x_{i}\mathfrak{m}(\mathbf{x})$, for $i=1,2,\ldots,n$, we may associate terms as in the proof of Proposition 4, using now (25), and we are done. ∎

5.3 The map $\phi_{\boldsymbol{\lambda}}$ preserves evaluations

In this subsection, we show that the $(\sigma,\delta^{\prime})$-evaluation of $F(\mathbf{x}+\boldsymbol{\lambda})\in\mathbb{F}[\mathbf{x};\sigma,\delta^{\prime}]$ at an affine point $\mathbf{a}\in\mathbb{F}^{n}$ coincides with the $(\sigma,\delta)$-evaluation of $F(\mathbf{x})\in\mathbb{F}[\mathbf{x};\sigma,\delta]$ at the affine point $\mathbf{a}+\boldsymbol{\lambda}\in\mathbb{F}^{n}$. See (3) in Section 2 for the definition of evaluation.

Proposition 10**.**

Let $\sigma:\mathbb{F}\longrightarrow\mathbb{F}^{n\times n}$ be a ring morphism and let $\delta,\delta^{\prime}:\mathbb{F}\longrightarrow\mathbb{F}^{n}$ be $\sigma$-derivations as in Definition 5, for a given $\boldsymbol{\lambda}\in\mathbb{F}^{n}$. The map $\phi_{\boldsymbol{\lambda}}:\mathbb{F}[\mathbf{x};\sigma,\delta]\longrightarrow\mathbb{F}[\mathbf{x};\sigma,\delta^{\prime}]$ in Definition 5 preserves evaluations over any point $\mathbf{a}\in\mathbb{F}^{n}$ after translation by the vector $\boldsymbol{\lambda}$. More concretely, according to the definition of evaluation in (3), it holds that

[TABLE]

for all $F(\mathbf{x})\in\mathbb{F}[\mathbf{x};\sigma,\delta]$ and all $\mathbf{a}\in\mathbb{F}^{n}$. In other words, the evaluation of $F(\mathbf{x}+\boldsymbol{\lambda})$ in the point $\mathbf{a}$ is $F(\mathbf{a}+\boldsymbol{\lambda})$.

Proof.

Analogous to the proof of Proposition 5. ∎

Again, using this result, we may extend $\phi_{\boldsymbol{\lambda}}$ to non-free multivariate skew polynomial rings, following the definitions in Section 2 and the notation in Corollary 1.

Corollary 3**.**

Let $\sigma:\mathbb{F}\longrightarrow\mathbb{F}^{n\times n}$ be a ring morphism and let $\delta,\delta^{\prime}:\mathbb{F}\longrightarrow\mathbb{F}^{n}$ be $\sigma$-derivations as in Definition 5, for a given $\boldsymbol{\lambda}\in\mathbb{F}^{n}$. It holds that

[TABLE]

Therefore, we may extend $\phi_{\boldsymbol{\lambda}}$ to a ring morphism between non-free multivariate skew polynomial rings, that is, to ring morphisms

[TABLE]

for any two-sided ideal $I\subseteq I^{\sigma,\delta}(\mathbb{F}^{n})$, where $J=\phi_{\boldsymbol{\lambda}}(I)\subseteq I^{\sigma,\delta^{\prime}}(\mathbb{F}^{n})$ is also a two-sided ideal. Furthermore, evaluations are also preserved by $\phi_{\boldsymbol{\lambda}}$ as in (18) over non-free multivariate skew polynomial rings.

Finally, the remarks on conjugacy and the product rule after Corollary 1 can be translated mutatis mutandis to translations of variables. We leave the details to the reader, but we remark that, in this case, $\mathbf{a}$ and $\mathbf{b}$ are $(\sigma,\delta^{\prime})$-conjugate if, and only if, $\mathbf{a}+\boldsymbol{\lambda}$ and $\mathbf{b}+\boldsymbol{\lambda}$ are $(\sigma,\delta)$-conjugate, with the same element $c\in\mathbb{F}^{*}$ in the conjugacy relation (20) in both cases.

5.4 The inverse of the map $\phi_{\boldsymbol{\lambda}}$

In this subsection, we show that the ring morphism $\phi_{\boldsymbol{\lambda}}$ has $\phi_{-\boldsymbol{\lambda}}$ as its inverse, and thus it is a ring isomorphism. The main result is the following.

Proposition 11**.**

Let $\sigma:\mathbb{F}\longrightarrow\mathbb{F}^{n\times n}$ be a ring morphism and let $\delta,\delta^{\prime},\delta^{\prime\prime}:\mathbb{F}\longrightarrow\mathbb{F}^{n}$ be $\sigma$-derivations such that

[TABLE]

for all $a\in\mathbb{F}$, for certain $\boldsymbol{\lambda},\boldsymbol{\lambda}^{\prime}\in\mathbb{F}^{n}$. It holds that

[TABLE]

In addition, we have that $\phi_{\mathbf{0}}={\rm Id}$, hence

[TABLE]

That is, $\phi_{\boldsymbol{\lambda}}^{-1}=\phi_{-\boldsymbol{\lambda}}$, and $\phi_{\boldsymbol{\lambda}}$ is a ring isomorphism.

Proof.

The proof follows the same lines as in the proof of Proposition 6. ∎

Remark 7**.**

Again, the results in this section hold automatically over non-free multivariate skew polynomial rings due to Corollary 3.

5.5 The map $\phi_{\boldsymbol{\lambda}}$ preserves degrees

As it was the case for linear transformation of variables, the map $\phi_{\boldsymbol{\lambda}}$ preserves degrees, which follows immediately from Proposition 11, analogously to the proof of Proposition 8.

Proposition 12**.**

Let $\sigma:\mathbb{F}\longrightarrow\mathbb{F}^{n\times n}$ be a ring morphism and let $\delta,\delta^{\prime}:\mathbb{F}\longrightarrow\mathbb{F}^{n}$ be $\sigma$-derivations as in Definition 5, for a given $\boldsymbol{\lambda}\in\mathbb{F}^{n}$. It holds that

[TABLE]

for all $F(\mathbf{x})\in\mathbb{F}[\mathbf{x};\sigma,\delta]$.

6 Affine transformations of variables

This section is devoted to defining general affine transformations of variables between the corresponding multivariate skew polynomial rings and showing that they are the only ring isomorphisms that preserve degrees.

We start with the following observation, whose proof is left to the reader.

Proposition 13**.**

Let $\sigma,\tau:\mathbb{F}\longrightarrow\mathbb{F}^{n\times n}$ be ring morphisms and let $\delta_{\sigma},\delta_{\tau}:\mathbb{F}\longrightarrow\mathbb{F}^{n}$ be a $\sigma$-derivation and a $\tau$-derivation, respectively, such that

[TABLE]

for all $a\in\mathbb{F}$, for some invertible matrix $A\in\mathbb{F}^{n\times n}$. Next, let $\delta_{\tau}^{\prime}:\mathbb{F}\longrightarrow\mathbb{F}^{n}$ be a $\tau$-derivation such that

[TABLE]

for all $a\in\mathbb{F}$, for some $\boldsymbol{\lambda}\in\mathbb{F}^{n}$. Then it holds that

[TABLE]

In other words, and using the notations in (16) and (24), we have that

[TABLE]

for all $F(\mathbf{x})\in\mathbb{F}[\mathbf{x};\sigma,\delta_{\sigma}]$, in the ring $\mathbb{F}[\mathbf{x};\tau,\delta_{\tau}^{\prime}]$.

Thus we deduce the following.

Corollary 4**.**

Any composition in any order of a finite collection of linear transformations and translations of variables over multivariate skew polynomial rings is the composition of one linear transformation and one translation.

This corollary motivates the following definition of affine transformations of variables over multivariate skew polynomial rings.

Definition 6**.**

Let $\sigma,\tau:\mathbb{F}\longrightarrow\mathbb{F}^{n\times n}$ and $\delta_{\sigma},\delta_{\tau},\delta_{\tau}^{\prime}:\mathbb{F}\longrightarrow\mathbb{F}^{n}$ be as in Proposition 13. An affine transformation of variables between the multivariate skew polynomial rings $\mathbb{F}[\mathbf{x};\sigma,\delta_{\sigma}]$ and $\mathbb{F}[\mathbf{x};\tau,\delta_{\tau}^{\prime}]$ is a ring isomorphism of the form

[TABLE]

Corollary 4 also implies that the composition of affine transformations is again an affine transformation. However, the domains and codomains of such maps are not all the same when $(\sigma,\delta_{\sigma})\neq({\rm Id},0)$ or $\mathbb{F}$ is not commutative.

To conclude, we show that affine transformations constitute all left $\mathbb{F}$-linear ring isomorphisms (thus left $\mathbb{F}$-algebra isomorphisms) between multivariate skew polynomial rings that preserve degrees.

Theorem 4**.**

Let $\sigma,\tau:\mathbb{F}\longrightarrow\mathbb{F}^{n\times n}$ be ring morphisms and let $\delta_{\sigma},\delta_{\tau}^{\prime}:\mathbb{F}\longrightarrow\mathbb{F}^{n}$ be a $\sigma$-derivation and a $\tau$-derivation, respectively. A map $\mathcal{T}:\mathbb{F}[\mathbf{x};\sigma,\delta_{\sigma}]\longrightarrow\mathbb{F}[\mathbf{x};\tau,\delta_{\tau}^{\prime}]$ is a left $\mathbb{F}$-linear ring isomorphism such that $\deg(\mathcal{T}(F(\mathbf{x})))=\deg(F(\mathbf{x}))$, for all $F(\mathbf{x})\in\mathbb{F}[\mathbf{x};\sigma,\delta_{\sigma}]$, if and only if, there exists a $\tau$-derivation $\delta_{\tau}:\mathbb{F}\longrightarrow\mathbb{F}^{n}$, an invertible matrix $A\in\mathbb{F}^{n\times n}$ and a vector $\boldsymbol{\lambda}\in\mathbb{F}^{n}$, all as in Proposition 13, such that $\mathcal{T}=\mathcal{T}_{A,\boldsymbol{\lambda}}$.

Proof.

The direct implication follows from Definition 6 and Propositions 4, 8, 9 and 12. For the reversed implication, the preservation of degrees for the monomials $x_{1},x_{2},\ldots,x_{n}$ implies that there exist an invertible matrix $A\in\mathbb{F}^{n\times n}$ and a vector $\boldsymbol{\lambda}\in\mathbb{F}^{n}$ such that

[TABLE]

Let $a\in\mathbb{F}$. In the ring $\mathbb{F}[\mathbf{x};\sigma,\delta_{\sigma}]$, we have that $\mathbf{x}a=\sigma(a)\mathbf{x}+\delta_{\sigma}(a)$ by (1). Hence

[TABLE]

Next, since $\mathcal{T}$ is a left $\mathbb{F}$-linear ring isomorphism, we also have that

[TABLE]

Comparing (27) and (28), the reader may check that there exists a $\tau$-derivation $\delta_{\tau}:\mathbb{F}\longrightarrow\mathbb{F}^{n}$, namely $\delta_{\tau}=A^{-1}\delta_{\sigma}$, such that $\sigma$, $\tau$, $\delta_{\sigma}$, $\delta_{\tau}$ and $\delta_{\tau}^{\prime}$ are all related as in Proposition 13 for the matrix $A\in\mathbb{F}^{n\times n}$ and the vector $\boldsymbol{\lambda}\in\mathbb{F}^{n}$.

Finally, proceeding recursively on monomials $\mathfrak{m}(\mathbf{x})\in\mathcal{M}$ as in the proofs of Propositions 4 and 9, the reader may check that $\mathcal{T}=\phi_{\boldsymbol{\lambda}}\circ\varphi_{A}=\mathcal{T}_{A,\boldsymbol{\lambda}}$, and we are done. ∎

7 Classification over finite fields

With all the tools gathered up to this point, we are ready to show that all multivariate skew polynomial rings over a finite field $\mathbb{F}_{q}$ are naturally isomorphic as $\mathbb{F}_{q}$-algebras to some multivariate skew polynomial ring of the form $\mathbb{F}_{q}[\mathbf{x};{\rm diag}(\sigma_{1},\sigma_{2},\ldots,\sigma_{n}),0]$ (Theorem 5). Even further, two such representations only differ in a permutation of the field automorphisms $\sigma_{1},\sigma_{2},\ldots,\sigma_{n}:\mathbb{F}_{q}\longrightarrow\mathbb{F}_{q}$, providing a full classification of multivariate skew polynomial rings over finite fields (Theorem 6).

For ease of notation, given field automorphisms $\sigma_{1},\sigma_{2},\ldots,\sigma_{n}:\mathbb{F}_{q}\longrightarrow\mathbb{F}_{q}$, we will denote

[TABLE]

The next result extends the discussion given in [5, Sec. 8.3] (see also [15, Prop. 40]) from the case $n=1$ to the general case.

Theorem 5**.**

Let $\sigma:\mathbb{F}_{q}\longrightarrow\mathbb{F}_{q}^{n\times n}$ be a ring morphism and let $\delta:\mathbb{F}_{q}\longrightarrow\mathbb{F}_{q}^{n}$ be a $\sigma$-derivation. By Theorem 2, there exist an invertible matrix $A\in\mathbb{F}_{q}^{n\times n}$ and field automorphisms $\sigma_{1},\sigma_{2},\ldots,\sigma_{n}:\mathbb{F}_{q}\longrightarrow\mathbb{F}_{q}$ such that

[TABLE]

for all $a\in\mathbb{F}_{q}$, where $\tau={\rm diag}(\sigma_{1},\sigma_{2},\ldots,\sigma_{n})$. Since the map $A^{-1}\delta$ is a $\tau$-derivation by Proposition 3, we deduce from Theorem 1 that there exists a vector $\boldsymbol{\lambda}\in\mathbb{F}_{q}^{n}$ such that

[TABLE]

for all $a\in\mathbb{F}_{q}$. Therefore, by the results in Sections 4 and 5, the affine transformation of variables (Section 6)

[TABLE]

is an $\mathbb{F}_{q}$-algebra isomorphism that preserves evaluations (see Propositions 5 and 10) and degrees (see Propositions 8 and 12) and that may naturally be extended to non-free multivariate skew polynomial rings (see Corollaries 1 and 3).

Theorem 5 says that we may simplify $\mathbb{F}_{q}[\mathbf{x};\sigma,\delta]$ to $\mathbb{F}_{q}[\mathbf{x};\sigma_{1},\sigma_{2},\ldots,\sigma_{n}]$ via an affine transformation of variables. The next theorem shows that two such reductions only differ by a permutation of $\sigma_{1},\sigma_{2},\ldots,\sigma_{n}$.

Theorem 6**.**

Let $\sigma_{1},\sigma_{2},\ldots,\sigma_{n},\tau_{1},\tau_{2},\ldots,\tau_{n}:\mathbb{F}_{q}\longrightarrow\mathbb{F}_{q}$ be field automorphisms. There exists an affine transformation

[TABLE]

if, and only if, there exists a bijection $s:\{1,2,\ldots,n\}\longrightarrow\{1,2,\ldots,n\}$ such that $\tau_{i}=\sigma_{s(i)}$, for $i=1,2,\ldots,n$.

Proof.

The reversed implication is trivial. We now prove the direct implication. If there exists such an affine transformation, then by Theorem 4, there exists an invertible matrix $A\in\mathbb{F}_{q}^{n\times n}$ such that

[TABLE]

where $c\in\mathbb{F}_{q}^{*}$ is a primitive element. Since the eigenvalues of a square matrix are invariant by similarity of matrices, we deduce that there exists a bijection $s:\{1,2,\ldots,n\}\longrightarrow\{1,2,\ldots,n\}$ such that $\tau_{i}(c)=\sigma_{s(i)}(c)$, for $i=1,2,\ldots,n$. Now, if $\tau_{i}(c)=\sigma_{s(i)}(c)$, then $\tau_{i}(a)=\sigma_{s(i)}(a)$, for all $a\in\mathbb{F}_{q}$, since $c\in\mathbb{F}_{q}^{*}$ is a primitive element, and we are done. ∎

In conclusion, if we identify multivariate skew polynomial rings that are isomorphic by an $\mathbb{F}_{q}$-algebra isomorphism that preserves degrees (i.e., by an affine transformation of variables), then such classes are represented by the rings $\mathbb{F}_{q}[\mathbf{x};\sigma_{1},\sigma_{2},\ldots,\sigma_{n}]$ modulo permutations of the field automorphisms $\sigma_{1},\sigma_{2},\ldots,\sigma_{n}$.

Finally, Theorems 1, 2 and 5 give an explicit method to find the reduced form $\mathbb{F}_{q}[\mathbf{x};\sigma_{1},\sigma_{2},\ldots,\sigma_{n}]$ from $\sigma(c)\in\mathbb{F}_{q}^{n\times n}$ and $\delta(c)\in\mathbb{F}_{q}^{n}$, for a primitive element $c\in\mathbb{F}_{q}^{*}$.

Acknowledgement

The author gratefully acknowledges the support from The Independent Research Fund Denmark (Grant No. DFF-7027-00053B).