Analysis of computing Gröbner bases and Gröbner degenerations via theory of signatures

Abstract

The signatures of polynomials were originally introduced by Faugère for the efficient computation of Gröbner bases [Fau02], and redefined by Arri-Perry [AP11] as the standard monomials modulo the module of syzygies. Since it is difficult to determine signatures, Vaccon-Yokoyama [VY17] introduced an alternative object called guessed signatures. In this paper, we consider a module $\mathrm{Gobs}(F)$ for a tuple of polynomials $F$ to analyse computation of Gröbner bases via theory of signatures. This is the residue module ${\mathrm{ini}}_{\prec }(\mathrm{Syz}(\mathrm{LM}(F)))/{\mathrm{ini}}_{\prec }(\mathrm{Syz}(F))$ defined by the initial modules of the syzygy modules with respect to the Schreyer order. We first show that $F$ is a Gröbner basis if and only if $\mathrm{Gobs}(F)$ is the zero module. Then we show that any homogeneous Gröbner basis with respect to a graded term order satisfying a common condition must contain the remainder of a reduction of an S-polynomial. We give computational examples of transitions of minimal free resolutions of $\mathrm{Gobs}(F)$ in a signature based algorithm. Finally, we show a connection between the module $\mathrm{Gobs}(F)$ and Gröbner degenerations.

25 pages