On the p-adic stability of the FGLM algorithm

TL;DR

This paper investigates the p-adic stability of the FGLM algorithm used for changing monomial orderings in Gröbner basis computations, providing bounds on precision loss and demonstrating stability through Smith Normal Form techniques.

Contribution

It proves the p-adic stability of FGLM and its variant for shape position, offering explicit bounds on precision loss during execution.

Findings

FGLM is p-adic stable with explicit precision loss bounds.

The variant for shape position is also stable.

Smith Normal Form is used to analyze stability.

Abstract

Nowadays, many strategies to solve polynomial systems use the computation of a Gr{\"o}bner basis for the graded reverse lexicographical ordering, followed by a change of ordering algorithm to obtain a Gr{\"o}bner basis for the lexicographical ordering. The change of ordering algorithm is crucial for these strategies. We study the p-adic stability of the main change of ordering algorithm, FGLM. We show that FGLM is stable and give explicit upper bound on the loss of precision occuring in its execution. The variant of FGLM designed to pass from the grevlex ordering to a Gr{\"o}bner basis in shape position is also stable. Our study relies on the application of Smith Normal Form computations for linear algebra.