Newton iteration for lexicographic Gr\"obner bases in two variables

TL;DR

This paper introduces a quadratic convergence Newton iteration method for computing lexicographic Gr"obner bases of zero-dimensional ideals in two variables, leveraging a structural syzygy result for explicit parameterization.

Contribution

It develops a novel Newton iteration approach that directly works with affine parameters of Gr"obner bases, improving computational efficiency for two-variable cases.

Findings

Quadratic convergence of the Newton iteration method.

Explicit parameterization of Gr"obner bases using syzygy structure.

Enhanced computational approach for two-variable zero-dimensional ideals.

Abstract

We present an $m$-adic Newton iteration with quadratic convergence for lexicographic Gr\"obner basis of zero dimensional ideals in two variables. We rely on a structural result about the syzygies in such a basis due to Conca and Valla, that allowed them to explicitly describe these Gr\"obner bases by affine parameters; our Newton iteration works directly with these parameters.