Classification of multivariate skew polynomial rings over finite fields via affine transformations of variables |
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| Institution: | 1. P.N. Lebedev Physical Institute, Russian Academy of Sciences, Leninskii Prospect 53, Moscow 119991, Russia;2. Moscow Institute of Physics and Technology (State University), Institutskii per. 9, Dolgoprudnyi, Moscow Region 141700, Russia;1. Department Mathematik, Friedrich-Alexander-Universität Erlangen-Nürnberg, Cauerstraße 11, 91058 Erlangen, Germany;2. Department of Mathematics, The Bishop''s School, La Jolla, CA 92037, United States of America;1. American University of Sharjah, PO Box 26666, Sharjah, United Arab Emirates;2. School of Mathematics and Statistics, Shandong University of Technology, Zibo, Shandong 255091, China;3. University of Scranton, Scranton, PA 18510, USA;1. University College, Yonsei University, 85 Songdogwahak-ro, Yeonsu-gu, Incheon, 21983, Republic of Korea;2. Department of Mathematics, Kangwon National University, 1 Gangwondaehakgil, Chuncheon, 24341, Republic of Korea;3. Department of Mathematics, Sogang University, 35 Baekbeom-ro, Seoul 04107, Republic of Korea;4. Department of Mathematics, Ewha Womans University, 11-1 Daehyun-Dong, Seodaemun-Gu, Seoul, 03760, Republic of Korea;1. National Key Laboratory of Science and Technology on Communications, University of Electronic Science and Technology of China, Chengdu 611731, China;2. Yangtze Delta Region Institute (Huzhou), University of Electronic Science and Technology of China, Huzhou 313001, China;3. Strategic Centre for Research on Privacy-Preserving Technologies and Systems, Nanyang Technological University, Singapore 637553, Singapore |
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| 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 on 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. |
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| Keywords: | Affine transformations Derivations Free polynomial rings Moore matrices Multivariate skew polynomial rings Vandermonde matrices |
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