2019/08/19 by Umberto Martínez-Peñas, Martínez-Peñas, Umberto
Computer Science · #11T06 #11T30 #12E10 #12E20 #Coding theory and cryptography #Cryptographic Implementations and Security #FOS: Mathematics #Network Packet Processing and Optimization #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.1908.06833
openalex publication_date 2019/08/19 · openalex created_date 2022/07/28 · openalex updated_date 2026/07/28
In this work, free multivariate skew polynomial rings are considered,\ntogether with their quotients over ideals of skew polynomials that vanish at\nevery point (which includes minimal multivariate skew polynomial rings). We\nprovide a full classification of such multivariate skew polynomial rings (free\nor not) over finite fields. To that end, we first show that all ring morphisms\nfrom the field to the ring of square matrices are diagonalizable, and that the\ncorresponding derivations are all inner derivations. Secondly, we show that all\nsuch multivariate skew polynomial rings over finite fields are isomorphic as\nalgebras to a multivariate skew polynomial ring whose ring morphism from the\nfield to the ring of square matrices is diagonal, and whose derivation is the\nzero derivation. Furthermore, we prove that two such representations only\ndiffer in a permutation of the field automorphisms appearing in the\ncorresponding diagonal. The algebra isomorphisms are given by affine\ntransformations of variables and preserve evaluations and degrees. In addition,\nours proofs show that the simplified form of multivariate skew polynomial rings\ncan be found computationally and explicitly.\n