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Normalization of Indexed Differentials by Extending Gröner Basis Theory

LIU Jiang, NI Feng, SONG Shihang, DU Mingjun   

  1. Department of Systems Science, University of Shanghai for Science and Technology, Shanghai 200093, China
  • Received:2020-12-01 Revised:2021-03-22 Online:2022-10-25 Published:2022-10-12
  • Supported by:
    This research was supported by the National Natural Science Foundation of China under Grant No.11701370

LIU Jiang, NI Feng, SONG Shihang, DU Mingjun. Normalization of Indexed Differentials by Extending Gröner Basis Theory[J]. Journal of Systems Science and Complexity, 2022, 35(5): 2016-2028.

It is a fundamental problem to determine the equivalence of indexed differential polynomials in both computer algebra and differential geometry.However,in the literature,there are no general computational theories for this problem.The main reasons are that the ideal generated by the basic syzygies cannot be finitely generated,and it involves eliminations of dummy indices and functions.This paper solves the problem by extending Göbner basis theory.The authors first present a division of the set of elementary indexed differential monomials$E_{\part}$/into disjoint subsets,by defining an equivalence relation on$E_{\part}$/based on Leibniz expansions of monomials.The equivalence relation on$E_{\part}$/also induces a division of a Göbner basis of basic syzygies into disjoint subsets.Furthermore,the authors prove that the dummy index numbers of the sim-monomials of the elements in each equivalence class of$E_{\part}$/have upper bounds,and use the upper bounds to construct fundamental restricted rings.Finally,the canonical form of an indexed differential polynomial proves to be the normal form with respect to a subset of the Göbner basis in the fundamental restricted ring.
[1] Liu J, Li H B, and Cao Y H, Simplification and normalization of indexed differentials involving coordinate transformation, Sci. China Ser. A, 2009, 52:2266-2286.
[2] Liu J, Normalization of indexed differentials based on function distance invariants, CASC 2017, Lecture Notes in Computer Science, 2017, 10490:285-300.
[3] Liu J and Ni F, Distance invariant method for normalization of indexed differentials, Journal of Symbolic Computation, 2020, 104:256-275.
[4] Liu J, An extension of Göbner basis theory to indexed polynomials without eliminations, Journal of Systems Science and Complexity, 2020, 33(5):1708-1718.
[5] Liu J, Normalization in Riemann tensor polynomial ring, Journal of Systems Science and Complexity, 2018, 31(2):569-580.
[6] Iima K and Yoshino Y, Göbner bases for the polynomial ring with infinite variables and their applications, Communications in Algebra, 2009, 37:3424-3437.
[7] Hillar C J and Sullivant S, Finite Göbner bases in infinite dimensional polynomial rings and applications, Advances in Mathematics, 2012, 229:1-25.
[8] Krone R, Equivariant Göbner bases of symmetric toric ideals, Proc. ISSAC'16, 2016, 311-318.
[9] Dung D H, Equivariant Göbner Bases, Master's thesis, Leiden University, 2010.
[10] Hillar C J, Kroner R, and Leykin A, Equivariant Göbner bases, 8th Mathematical Society of Japan Seasonal Institute Conference on Current Trends on Göbner Bases:50th Anniversary of Göbner Bases, Advanced Studies in Pure Mathematics, 2018, 77:129-154.
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