$p$-adic algorithm for bivariate Gr\"obner bases

Abstract

We present a $p$-adic algorithm to recover the lexicographic Gr\"obner basis $\mathcal G$ of an ideal in $\mathbb Q[x,y]$ with a generating set in $\mathbb Z[x,y]$, with a complexity that is less than cubic in terms of the dimension of $\mathbb Q[x,y]/\langle \mathcal G \rangle$ and softly linear in the height of its coefficients. We observe that previous results of Lazard's that use Hermite normal forms to compute Gr\"obner bases for ideals with two generators can be generalized to a set of $t\in \mathbb N^+$ generators. We use this result to obtain a bound on the height of the coefficients of $\mathcal G$, and to control the probability of choosing a \textit{good} prime $p$ to build the $p$-adic expansion of $\mathcal G$.