{"language":[{"iso":"eng"}],"date_updated":"2024-04-11T12:50:48Z","title":"Polynomials vanishing at lattice points in a convex set","publication":"arXiv:2107.05353","abstract":[{"text":"Let $P$ be a bounded convex subset of $\\mathbb R^n$ of positive volume.\r\nDenote the smallest degree of a polynomial $p(X_1,\\dots,X_n)$ vanishing on\r\n$P\\cap\\mathbb Z^n$ by $r_P$ and denote the smallest number $u\\geq0$ such that\r\nevery function on $P\\cap\\mathbb Z^n$ can be interpolated by a polynomial of\r\ndegree at most $u$ by $s_P$. We show that the values $(r_{d\\cdot P}-1)/d$ and\r\n$s_{d\\cdot P}/d$ for dilates $d\\cdot P$ converge from below to some numbers\r\n$v_P,w_P>0$ as $d$ goes to infinity. The limits satisfy $v_P^{n-1}w_P \\leq\r\nn!\\cdot\\operatorname{vol}(P)$. When $P$ is a triangle in the plane, we show\r\nequality: $v_Pw_P = 2\\operatorname{vol}(P)$. These results are obtained by\r\nlooking at the set of standard monomials of the vanishing ideal of $d\\cdot\r\nP\\cap\\mathbb Z^n$ and by applying the Bernstein--Kushnirenko theorem. Finally,\r\nwe study irreducible Laurent polynomials that vanish with large multiplicity at\r\na point. This work is inspired by questions about Seshadri constants.","lang":"eng"}],"extern":"1","date_created":"2024-04-11T12:43:04Z","external_id":{"arxiv":["2107.05353"]},"_id":"53420","status":"public","year":"2021","author":[{"last_name":"Gundlach","full_name":"Gundlach, Fabian","id":"100450","first_name":"Fabian"}],"user_id":"100450","type":"preprint","citation":{"chicago":"Gundlach, Fabian. “Polynomials Vanishing at Lattice Points in a Convex Set.” <i>ArXiv:2107.05353</i>, 2021.","mla":"Gundlach, Fabian. “Polynomials Vanishing at Lattice Points in a Convex Set.” <i>ArXiv:2107.05353</i>, 2021.","apa":"Gundlach, F. (2021). Polynomials vanishing at lattice points in a convex set. In <i>arXiv:2107.05353</i>.","short":"F. Gundlach, ArXiv:2107.05353 (2021).","ieee":"F. Gundlach, “Polynomials vanishing at lattice points in a convex set,” <i>arXiv:2107.05353</i>. 2021.","bibtex":"@article{Gundlach_2021, title={Polynomials vanishing at lattice points in a convex set}, journal={arXiv:2107.05353}, author={Gundlach, Fabian}, year={2021} }","ama":"Gundlach F. Polynomials vanishing at lattice points in a convex set. <i>arXiv:210705353</i>. Published online 2021."}}