Polynomials vanishing at lattice points in a convex set

TL;DR

This paper investigates the asymptotic behavior of polynomials vanishing on lattice points within convex sets, establishing limits related to volume and exploring connections to algebraic geometry and Seshadri constants.

Contribution

It introduces the limits of minimal polynomial degrees and interpolation degrees for dilated convex sets, linking these to volume and standard monomials, with exact results for planar triangles.

Findings

Limits of polynomial degrees converge to positive constants.

For triangles, the product of limits equals twice the area.

Inequality relating volume and polynomial invariants is established.

Abstract

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.