Newton iteration for lexicographic Gröbner bases in two variables

Abstract

We present an m-adic Newton iteration with quadratic convergence for lexicographic Gröbner 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 Gröbner bases by affine parameters; our Newton iteration works directly with these parameters.

Bivariate polynomial reduction and elimination ideal over finite fields

“…Following the dichotomic scheme of van der Hoeven and Larrieu, 2018, an efficient reduction algorithm modulo a bivariate lexicographic Gröbner basis is given in (Schost and St-Pierre, 2023a). Finally, although it retains certain assumptions about the ideal, we should also mention the multiplication bound Õ((de) 1.5 ) of (Hyun et al, 2019, Sec.…”