On the Bit-Size of Non-radical Triangular Sets

Dahan, Xavier

doi:10.1007/978-3-319-72453-9_19

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“…Evidences that a broad class of non-radical lexGbs also holds a factorization pattern are coming out. A key step in that direction lies in the development of CRT-based algorithms that reconstruct a lexGb from its primary components [11].…”

Section: Discussionmentioning

confidence: 99%

Lexicographic Groebner bases of bivariate polynomials modulo a univariate one

Dahan

2020

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Let T (x) ∈ k[x] be a monic non-constant polynomial and write R = k[x]/ T the quotient ring. Consider two bivariate polynomials a(x, y), b(x, y) ∈ R [y]. In a first part, T = p e is assumed to be the power of an irreducible polynomial p. A new algorithm that computes a minimal lexicographic Gröbner basis of the ideal a, b, p e , is introduced. A second part extends this algorithm when T is general through the "local/global" principle realized by a generalization of "dynamic evaluation", restricted so far to a polynomial T that is squarefree. The algorithm produces splittings according to the case distinction "invertible/nilpotent", extending the usual "invertible/zero" in classic dynamic evaluation. This algorithm belongs to the Euclidean family, the core being a subresultant sequence of a and b modulo T . In particular no factorization or Gröbner basis computations are necessary. The theoretical background relies on the Lazard's structural theorem for lexicographic Gröbner bases in two variables. An implementation is realized in Magma, and extensive benchmarks are provided that show clearly the benefits, sometimes huge, of this approach compared to the computation of Gröbner bases.

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Section: Discussionmentioning

confidence: 99%