Lexicographic Groebner bases of bivariate polynomials modulo a univariate one

TL;DR

This paper introduces a new algorithm for computing lexicographic Groebner bases of bivariate polynomials modulo a univariate polynomial, extending to general T with a local/global principle, avoiding factorization and improving efficiency.

Contribution

The paper presents a novel algorithm for minimal lexicographic Groebner bases computation that handles general T without factorization, using a local/global approach and subresultant sequences.

Findings

Algorithm avoids factorization and Groebner basis computations.

Benchmarks demonstrate significant efficiency improvements.

Extends dynamic evaluation to non-squarefree polynomials.

Abstract

Let T(x) in 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) in 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 Groebner 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 Groebner basis computations are necessary. The theoretical background relies on Lazard's structural theorem for lexicographic Groebner bases in two variables. An implementation is realized in Magma. Benchmarks show clearly the benefit, sometimes important, of this approach compared to the Groebner bases approach.