Newton iteration for lexicographic Gröbner bases in two variables

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.