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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 $u\geq0$ such thatevery function on $P\cap\mathbb Z^n$ can be interpolated by a polynomial ofdegree at most $u$ by $s_P$. We show that the values $(r_{d\cdot P}-1)/d$ and$s_{d\cdot P}/d$ for dilates $d\cdot P$ converge from below to some numbers$v_P,w_P>0$ as $d$ goes to infinity. The limits satisfy $v_P^{n-1}w_P \leqn!\cdot\operatorname{vol}(P)$. When $P$ is a triangle in the plane, we showequality: $v_Pw_P = 2\operatorname{vol}(P)$. These results are obtained bylooking at the set of standard monomials of the vanishing ideal of $d\cdotP\cap\mathbb Z^n$ and by applying the Bernstein--Kushnirenko theorem. Finally,we study irreducible Laurent polynomials that vanish with large multiplicity ata point. This work is inspired by questions about Seshadri constants.

Polynomials vanishing at lattice points in a convex set