An m-adic algorithm for bivariate Gröbner bases

Abstract

Let $A$ be a domain, with $m\subseteq A$ a maximal ideal, and let $F\subseteq A[x,y]$ be any finite generating set of an ideal with finitely many roots (in an algebraic closure of the fraction field $K$ of $A$). We present a randomized $m$-adic algorithm to recover the lexicographic Gröbner basis $G$ of $〈F〉\subseteq K[x,y]$, or of its primary component at the origin. We observe that previous results of Lazard's that use Hermite normal forms to compute Gröbner bases for ideals with two generators can be generalized to a generating set $F$ of cardinality greater than two. We use this result to bound the size of the coefficients of $G$, and to control the probability of choosing a good maximal ideal $m\subseteq A$. We give a complete cost analysis over number fields ($K=Q\left(\alpha \right)$) and function fields (

), and we obtain a complexity that is less than cubic in terms of the dimension of $K/〈G〉$ and softly linear in the size of its coefficients.