  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 $\mathcal G$.
