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 $\mathcal G$.

翻译：我们提出一种$p$-adic算法，用于恢复$\mathbb Q[x,y]$中由$\mathbb Z[x,y]$生成集给出的理想的字典序Gröbner基$\mathcal G$。该算法的复杂度关于$\mathbb Q[x,y]/\langle \mathcal G \rangle$的维数为低于三次，且关于其系数的高度为软线性。我们观察到，Lazard先前利用Hermite标准型计算两个生成元理想Gröbner基的结果可推广至$t\in \mathbb N^+$个生成元的集合。我们利用此结果获得$\mathcal G$系数高度的界，并控制选择构造$\mathcal G$的$p$-adic展开所需的\emph{优}素数$p$的概率。