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Theory and Applications of Linearized Multivariate Skew Polynomials

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Theory and Applications of Linearized Multivariate Skew Polynomials Theory and applications of linearized multivariate skew polynomials Umberto Mart´ınez-Pe˜nas ∗ Institute of Computer Science and Mathematics, University of Neuchˆatel, Switzerland Abstract In this work, linearized multivariate skew polynomials with coefficients over division rings are introduced, which generalize univariate linearized polynomials, group algebras of finite groups of division ring automorphisms, and algebras of derivations, among others. It is shown that they are right linear over a division subring called centralizer, and their natural evaluation is connected to the remainder-based evaluation of free multivariate skew polynomials. It is shown that P-independence corresponds to right linear independence over centralizers of pair-wise disjoint conjugacy classes. Hence it is deduced that finitely generated P-closed sets correspond to lists of finite-dimensional right vector spaces according to the partition of the P-closed set into conjugacy classes. Such finitely generated P-closed sets are the sets where Lagrange interpolation works as expected. It is also shown that products of free multivariate skew polynomials translate into coordinate-wise compositions (one per conjugacy class) of linearized multivariate skew polynomials, and compositions over a single conjugacy class translate into matrix products over the corresponding centralizers. Several applications of these results are given. First, linearized multivariate Vander- monde matrices are introduced, which generalize multivariate Moore and Wronskian matri- ces. The previous results give an explicit method to determine their ranks in general. Such matrices also give rise to a novel version of skew and linearized Reed-Muller codes, which are introduced and connected in this work. Finally, we introduce P-Galois extensions of division rings by considering the subring to be a centralizer corresponding to a conjugacy class given by a free multivariate skew polynomial ring. Such extensions generalize (finite) Galois extension of fields. Three Galois-theoretic results are generalized to such extensions: Artin’s theorem on the dimension of the larger division ring over its subring, the Galois correspondence, and Hilbert’s Theorem 90. Keywords: Galois theory, Hilbert’s Theorem 90, Lagrange interpolation, linearized polynomials, Moore matrices, Reed-Muller codes, skew polynomials, Vandermonde matrices, Wronskian matrices. MSC: 12E10, 12E15, 12F10, 16S36, 94B60. arXiv:2001.01273v2 [math.RA] 17 Sep 2020 1 Introduction The concept of univariate skew polynomial was introduced by Ore in [38]. They are defined as elements of the most general polynomial ring where the commutativity axiom is dropped, that is, a (commutative or non-commutative) left algebra over a division ring with a left basis of ∗[email protected] 1 monomials 1, x, x2,... whose product satisfies that xi + xj = xi+j , for all non-negative integers i, j, and such that the degree of the product of two skew polynomials is the sum of their degrees. Here we are slightly bending the usual notion of algebra [8]. By left algebra, we mean a left vector space with a ring structure whose product is linear on the first component (rather than bilinear as in the commutative case). A natural definition of evaluation of univariate skew polynomials, via Euclidean division, was introduced by Lam and Leroy in the works [22, 25]. Thanks to this concept of evaluation, Lam and Leroy introduced the concept of P-independence of evaluation points in [22, 24], which in turn gives rise to the concept of P-closed set (Definition 20), and P-basis (Definition 23) of a P-closed set. Intuitively, a finite set of evaluation points is P-independent if we may perform Lagrange interpolation over them, i.e., any set of values (of the right size) can be attained by evaluating some skew polynomial over such evaluation points (see Theorem 2). In [22, Theorem 23], it was shown that a set of evaluation points is P-independent if, and only if, the subsets in its partition into conjugacy classes (Definition 18) are each P-independent. Later in [25, Theorem 4.5], it was shown that a set of evaluation points, all from the same conjugacy class, is P-independent if, and only if, the exponents in the conjugacy relation are right linearly independent over the corresponding centralizer (Definition 10). With these two results, Lam and Leroy gave a simple explicit method to find the rank of matrices obtained by evaluating (univariate) skew polynomials, which generalize Vandermonde matrices [43] and are related to Moore matrices [33] and Wronskian matrices [15]. Later on, such method for finding the rank of such general Vandermonde matrices that use evaluations of skew polynomials was used in [29] to show that linearized Reed-Solomon codes (introduced in [29, Definition 31]) have maximum possible minimum sum-rank distance. Linearized Reed-Solomon codes are defined by evaluating certain operator polynomials that generalize classical univariate linearized polynomials over finite fields [28, Chapter 3]. Evaluations of such polynomials are tightly connected to Lam and Leroy’s concept of evaluation for skew polynomials via a particu- lar case of a result by Leroy [27, Theorem 2.8]. Since such operator polynomials are right linear over the corresponding centralizer, they can be seen as a linearization of skew polynomials. The generator matrices of linearized Reed-Solomon codes [29, page 604] are a linearized version of the skew Vandermonde matrices defined in [22, 25] and simultaneously recover as particular cases Vandermonde, Moore and Wronskian matrices. For this reason, linearized Reed-Solomon codes also recover as particular cases Reed-Solomon codes [40], which are MDS (maximum distance separable) and Gabidulin codes [10], which are MRD (maximum rank distance). These codes have numerous applications in error correction in telecommunications, repair in data storage or information-theoretical security, among others. Most notably, Reed-Solomon codes been ex- tensively used in practice, including CDs, DVDs, QR codes, satellite communications and the storage system RAID 6. In [31], free multivariate skew polynomials were introduced, following Ore’s definition: They are the most general polynomial ring in several free variables (variables are not allowed to com- mute with each other) such that the product of two monomials consists in appending them and the degree of a product of two skew polynomials is the sum of their degrees. Thanks to the lack of relations between the variables, the concept of evaluation was extended in [31, Definition 9] due to the uniqueness of remainders in the Euclidean division [31, Lemma 5], which cannot be guaranteed for iterated skew polynomial rings (see [31, Remark 8] and [12, Example 3.7]) or if the variables are allowed to commute (see [31, Remark 7]). The concepts of conjugacy, P-independence and skew Vandermonde matrices were then extended to such a multivariate case in [31], leading to a skew Lagrange interpolation theorem [31, Theorem 4] and equating the rank of a skew Vandermonde matrix to the rank of the P-closed set generated by the corresponding evaluation points [31, Proposition 41]. 2 In this work, we introduce a concept of multivariate polynomials on certain operators, as done in [29], which we will call linearized multivariate skew polynomials (Subsection 2.2), and we show that their natural evaluation is also tightly connected to the arithmetic evaluation (that is, based on Euclidean divisions) of free multivariate skew polynomials (Subsection 2.3). We will use this connection and skew Lagrange interpolation [31, Theorem 4] to extend the important results [22, Theorem 23] and [25, Theorem 4.5] to the multivariate case, finding an explicit representation of P-closed sets as a disjoint union or a list of right vector spaces over the corresponding centralizers (Theorems 4 and 5 in Section 3), and similarly for their P-bases. We will then show that compositions of linearized multivariate skew polynomials, seen as right linear maps, coincide with matrix products, and products of free multivariate skew polynomials can be mapped onto coordinate-wise compositions of linearized multivariate skew polynomials over pair-wise disjoint conjugacy classes (Section 4), which is hence equivalent to products of block-diagonal matrices. As a consequence, we deduce in Corollary 45 that quotients of free multivariate skew polynomial rings over the ideal of skew polynomials vanishing on a finite union of finitely generated conjugacy classes are semisimple rings [23, Definition (2.5)]. Moreover, they are simple rings [23, Definition (2.1)] in the case of a single conjugacy class (Corollary 44). We note that all of these results particularize to non-trivial results on the free conventional multivariate polynomial ring (where variables do not commute with each other but commute with constants) over an arbitrary division ring. The case where such division rings are fields (i.e., commutative) was extensively studied in [8], and the case of arbitrary division rings but where variables commute with each other was studied in [2]. However, our results in the general case are new to the best of our knowledge. The final two sections of this work constitute applications of the theory developed up to this point. In Subsection 5.2, we will define linearized multivariate Vandermonde matrices, connect them to skew multivariate Vandermonde matrices [31], and provide a simple explicit criterion to de- termine their ranks (Theorem 11 in Subsection 5.2) similar to that obtained by combining [22, Theorem 23] and [25, Theorem 4.5] in the univariate case. In Subsection 5.3, we introduce skew and linearized Reed-Muller codes, calculate their dimension and show a connection between their minimum skew and sum-rank distances (respectively) and their minimum Hamming distance. As we will show, skew Reed-Muller are similar but not exactly the same as those introduced in [12], and linearized Reed-Muller codes recover the version of Reed-Muller codes in [4] as the particular case of a single conjugacy class. Finally, in Section 6 we introduce the concept of P-Galois extensions of division rings, which generalize Galois extensions of fields.
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