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.

翻译：我们提出了一种具有二次收敛性的$m$进牛顿迭代法，用于求解双变量零维理想的字典序Gröbner基。该方法基于Conca与Valla关于此类基中合冲结构的研究成果，该成果允许他们通过仿射参数显式描述这些Gröbner基；我们的牛顿迭代法直接作用于这些参数。