preprint Fabian Gundlach author 100450 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 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 $(r_{d\cdot P}-1)/d$ 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 $v_P^{n-1}w_P \leq n!\cdot\operatorname{vol}(P)$. When $P$ is a triangle in the plane, we show equality: $v_Pw_P = 2\operatorname{vol}(P)$. 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. 2021 eng 2107.05353 yes Gundlach, Fabian. “Polynomials Vanishing at Lattice Points in a Convex Set.” <i>ArXiv:2107.05353</i>, 2021. F. Gundlach, ArXiv:2107.05353 (2021). F. Gundlach, “Polynomials vanishing at lattice points in a convex set,” <i>arXiv:2107.05353</i>. 2021. Gundlach, F. (2021). Polynomials vanishing at lattice points in a convex set. In <i>arXiv:2107.05353</i>. @article{Gundlach_2021, title={Polynomials vanishing at lattice points in a convex set}, journal={arXiv:2107.05353}, author={Gundlach, Fabian}, year={2021} } Gundlach F. Polynomials vanishing at lattice points in a convex set. <i>arXiv:210705353</i>. Published online 2021. Gundlach, Fabian. “Polynomials Vanishing at Lattice Points in a Convex Set.” <i>ArXiv:2107.05353</i>, 2021. 534202024-04-11T12:43:04Z2024-04-11T12:50:48Z