P000030

*姜 刘 (上海理工大学)

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 Gr\"obner basis theory. We first present a division of the set of elementary indexed differential monomials $\mathbf{E}_{\part}$ into disjoint subsets, by defining an equivalence relation on $\mathbf{E}_{\part}$ based on Leibniz expansions of monomials. The equivalence relation on $\mathbf{E}_{\part}$ also induces a division of a Gr\"obner basis of basic syzygies into disjoint subsets. Furthermore, we prove that the dummy index numbers of the sim-monomials of the elements in each equivalence class of $\mathbf{E}_{\part}$ 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 Gr\"obner basis in the fundamental restricted ring. In further work, the results of this paper will be used to develop computational theories for more general indexed polynomials, and for similar-indexed polynomials, which can be applied to mechanical theorem-proving in differential geometry and to solving some basic problems in graph theory, including finding the canonical labeling of multi-weighted directed graphs, and the problem of how to add the least edges to turn a directed graph into a Hamiltonian graph.

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