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A comparison of unrestricted dynamic Gröbner Basis algorithms

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Applicable Algebra in Engineering, Communication and Computing Aims and scope

Abstract

Dynamic Gröbner Basis algorithms aim to obtain small reduced Gröbner Bases in terms of number of polynomials and monomials. Dynamic algorithms allowing previous leading monomials to change during the execution, called unrestricted algorithms, are underexplored in the literature and the only previous unrestricted algorithm was not implemented. In this paper, we introduce a definition of neighborhoods for monomial orderings, prove it is well-behaved with respect to the commonly used Hilbert heuristic, propose new unrestricted algorithms based on these neighborhoods and compare them experimentally to some previous dynamic algorithms. Our results show that simple unrestricted algorithms based on random sampling often lead to small output bases containing polynomials of low degree, with little overhead.

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Acknowledgements

The author would like to thank Marcus Ritt and John Perry for many relevant discussions on the subject and the anonymous referees for their suggestions. This work was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001 and by Conselho Nacional de Desenvolvimento Científico (CNPq).

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  1. Instituto de Informática, UFRGS, Caixa postal 15064 Av. Bento Gonçalves, 9500, Porto Alegre, RS, 91501-970, Brazil

    Gabriel Mattos Langeloh

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Correspondence to Gabriel Mattos Langeloh.

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Langeloh, G.M. A comparison of unrestricted dynamic Gröbner Basis algorithms. AAECC 31, 389–409 (2020). https://doi.org/10.1007/s00200-020-00449-5

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