Polynomials vanishing at lattice points in a convex set

Let $P$ be a bounded convex subset of ${\mathbb{R}}^n$ of positive volume. Denote the smallest degree of a polynomial $p(X_1,\dots ,X_n)$ vanishing on $P\cap {\mathbb{Z}}^n$ by $r_P$ and denote the smallest number $u\ge 0$ such that every function on $P\cap {\mathbb{Z}}^n$ can be interpolated by a polynomial of degree at most $u$ by $s_P$. We show that the values $(r_{d\cdot P}-1)\mathrm{/}d$ and $s_{d\cdot P}\mathrm{/}d$ for dilates $d\cdot P$ converge from below to some numbers $v_P,w_P>0$ as $d$ goes to infinity. The limits satisfy $v_P^{n-1}w_P \leq n!\cdot\operatorname{vol}(P)$\mathrm{.}When$P$ is a triangle in the plane, we show equality: $v_P{w}_P=2\mathrm{vol}(P)$. These results are obtained by looking at the set of standard monomials of the vanishing ideal of $d\cdot P\cap\mathbb Z^n$ and by applying the Bernstein–Kushnirenko theorem. Finally, we study irreducible Laurent polynomials that vanish with large multiplicity at a point. This work is inspired by questions about Seshadri constants.