Normalization of Indexed Differentials by Extending Gröbner Basis Theory

Abstract

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 Gröbner basis theory. The authors first present a division of the set of elementary indexed differential monomials $E_{\partial\kern-0.35em\raise0.22ex\hbox{/}}$ into disjoint subsets, by defining an equivalence relation on $E_{\partial\kern-0.35em\raise0.22ex\hbox{/}}$ based on Leibniz expansions of monomials. The equivalence relation on $E_{\partial\kern-0.35em\raise0.22ex\hbox{/}}$ also induces a division of a Gröbner 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 $E_{\partial\kern-0.35em\raise0.22ex\hbox{/}}$ 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 Gröbner basis in the fundamental restricted ring.