Ivan Soprunovhttps://works.bepress.com/ivan-soprunov/Recent works by Ivan Soprunoven-usCopyright (c) 2019 All rights reserved.Tue, 01 May 2018 07:00:00 +00003600Generalized Multiplicities of Edge Idealshttps://works.bepress.com/ivan-soprunov/8/We explore connections between the generalized multiplicities of square-free monomial ideals and the combinatorial structure of the underlying hypergraphs using methods of commutative algebra and polyhedral geometry. For instance, we show that the j-multiplicity is multiplicative over the connected components of a hypergraph, and we explicitly relate the j-multiplicity of the edge ideal of a properly connected uniform hypergraph to the Hilbert–Samuel multiplicity of its special fiber ring. In addition, we provide general bounds for the generalized multiplicities of the edge ideals and compute these invariants for classes of uniform hypergraphs.Tue, 01 May 2018 07:00:00 +0000https://works.bepress.com/ivan-soprunov/8/ArticlesWulff Shapes and a Characterization of Simplices via a Bezout Type Inequalityhttps://works.bepress.com/ivan-soprunov/12/<p>Inspired by a fundamental theorem of Bernstein, Kushnirenko, and Khovanskii we study the following Bezout type inequality for mixed volumes <em><strong>V</strong>(<strong>L</strong><sub>1</sub>,...,<strong>L</strong><sub>n</sub>)<strong>V</strong><sub>n</sub>(<strong>K</strong>) ≤ <strong>V</strong>(<strong>L</strong><sub>1</sub>,<strong>K</strong>[n-1])<strong>V</strong>(<strong>L</strong><sub>2</sub>,..., <strong>L</strong><sub>{n}</sub>,<strong>K</strong>)</em>. We show that the above inequality characterizes simplices, i.e. if<em> <strong>K</strong></em> is a convex body satisfying the inequality for all convex bodies<strong><em> L</em></strong><sub>1</sub>, ...,<em><strong> L</strong><sub>n</sub></em> ⊂ R<em><sup>n</sup></em>, then <strong><em>K</em></strong> must be an <em>n</em>-dimensional simplex. The main idea of the proof is to study perturbations given by Wulff shapes. In particular, we prove a new theorem on differentiability of the support function of the Wulff shape, which is of independent interest. In addition, we study the Bezout inequality for mixed volumes introduced in arXiv:1507.00765 . We introduce the class of weakly decomposable convex bodies which is strictly larger than the set of all polytopes that are non-simplices. We show that the Bezout inequality in arXiv:1507.00765 characterizes weakly indecomposable convex bodies.</p>
Mon, 08 Jan 2018 08:00:00 +0000https://works.bepress.com/ivan-soprunov/12/ArticlesLet Me Tell You My Favorite Lattice-point Problem. . .https://works.bepress.com/ivan-soprunov/18/<p>This collection was compiled by Bruce Reznick from problems presented at the 2006 AMS/IMS/SIAM Summer Research Conference on Integer points in polytopes. SupposeP Rd is a convex rational d-polyhedron. The solid angle !P(x) of a point x (with respect toP) is a real number equal to the proportion of a small ball centered at x that is contained inP. That is, we let B (x) denote the ball of radius centered at x and dene !P(x) := vol (B (x)\P) volB (x) for all positive suciently small. We note that when x = 2P, !P(x) = 0; when x2P , !P(x) = 1; when x2 @P, 0 < !P(x) < 1. We dene</p>
Mon, 01 Jan 2018 08:00:00 +0000https://works.bepress.com/ivan-soprunov/18/ArticlesCriteria for Strict Monotonicity of the Mixed Volume of Convex Polytopeshttps://works.bepress.com/ivan-soprunov/9/<p>Let P<sub>1</sub>,..., P<sub>n</sub> and Q<sub>1,...,</sub>Q<sub>n</sub> be convex polytopes in <strong>R</strong><sup>n</sup> such that P<sub>i</sub> is a proper subset of Q<sub>i</sub> . It is well-known that the mixed volume has the monotonicity property: V (P1,...,P<sub>n</sub>) is less than or equal to V (Q<sub>1,...,</sub>Q<sub>n</sub>) . We give two criteria for when this inequality is strict in terms of essential collections of faces as well as mixed polyhedral subdivisions. This geometric result allows us to characterize sparse polynomial systems with Newton polytopes P<sub>1</sub>,..., P<sub>n</sub> whose number of isolated solutions equals the normalized volume of the convex hull of P<sub>1 </sub>U...U P<sub>n</sub> . In addition, we obtain an analog of Cramer's rule for sparse polynomial systems.</p>
Fri, 24 Feb 2017 08:00:00 +0000https://works.bepress.com/ivan-soprunov/9/ArticlesBezout Inequality for Mixed Volumeshttps://works.bepress.com/ivan-soprunov/3/In this paper we consider the following analog of Bezout inequality for mixed volumes: V(P1,…,Pr,Δn−r)Vn(Δ)r−1≤∏i=1rV(Pi,Δn−1) for 2≤r≤n. We show that the above inequality is true when Δ is an n-dimensional simplex and P1,…,Pr are convex bodies in Rn. We conjecture that if the above inequality is true for all convex bodies P1,…,Pr, then Δ must be an n-dimensional simplex. We prove that if the above inequality is true for all convex bodies P1,…,Pr, then Δ must be indecomposable (i.e. cannot be written as the Minkowski sum of two convex bodies which are not homothetic to Δ), which confirms the conjecture when Δ is a simple polytope and in the 2-dimensional case. Finally, we connect the inequality to an inequality on the volume of orthogonal projections of convex bodies as well as prove an isomorphic version of the inequality.Thu, 01 Dec 2016 08:00:00 +0000https://works.bepress.com/ivan-soprunov/3/ArticlesEventual Quasi-Linearity of The Minkowski Lengthhttps://works.bepress.com/ivan-soprunov/6/<p>The Minkowski length of a lattice polytope PP is a natural generalization of the lattice diameter of PP. It can be defined as the largest number of lattice segments whose Minkowski sum is contained in PP. The famous Ehrhart theorem states that the number of lattice points in the positive integer dilates tPtP of a lattice polytope PP behaves polynomially in t∈Nt∈N. In this paper we prove that for any lattice polytope PP, the Minkowski length of tPtP for t∈Nt∈N is eventually a quasi-polynomial with linear constituents. We also give a formula for the Minkowski length of coordinates boxes, degree one polytopes, and dilates of unimodular simplices. In addition, we give a new bound for the Minkowski length of lattice polygons and show that the Minkowski length of a lattice triangle coincides with its lattice diameter.</p>
Tue, 01 Nov 2016 07:00:00 +0000https://works.bepress.com/ivan-soprunov/6/ArticlesMultigraded Hilbert Functions and Toric Complete Intersection Codeshttps://works.bepress.com/ivan-soprunov/4/Let X be a complete n-dimensional simplicial toric variety with homogeneous coordinate ring S . We study the multigraded Hilbert function HYHY of reduced 0-dimensional subschemes Y in X . We provide explicit formulas and prove non-decreasing and stabilization properties of HYHY when Y is a 0-dimensional complete intersection in X. We apply our results to computing the dimension of some evaluation codes on 0-dimensional complete intersection in simplicial toric varieties.Mon, 01 Aug 2016 07:00:00 +0000https://works.bepress.com/ivan-soprunov/4/ArticlesCharacterization of Simplices via the Bezout Inequality for Mixed Volumeshttps://works.bepress.com/ivan-soprunov/14/<p>We consider the following Bezout inequality for mixed volumes: V (K1, . . . ,Kr, Δ[n − r])Vn(Δ)r−1 ≤ r i=1 V (Ki, Δ[n − 1]) for 2 ≤ r ≤ n. It was shown previously that the inequality is true for any -dimensional simplex and any convex bodies in . It was conjectured that simplices are the only convex bodies for which the inequality holds for arbitrary bodies in . In this paper we prove that this is indeed the case if we assume that is a convex polytope. Thus the Bezout inequality characterizes simplices in the class of convex -polytopes. In addition, we show that if a body satisfies the Bezout inequality for all bodies , then the boundary of cannot have points not lying in a boundary segment. In particular, it cannot have points with positive Gaussian curvature.</p>
Fri, 10 Jun 2016 07:00:00 +0000https://works.bepress.com/ivan-soprunov/14/ArticlesTropical Determinant on Transportation Polytopeshttps://works.bepress.com/ivan-soprunov/11/<p>Let D<sup>k,l</sup>(m, n)be the set of all the integer points in the transportation polytope of kn × ln matrices with row sums lm and column sums km. In this paper we find the sharp lower bound on the tropical determinant over the set D<sup>k,l</sup>(m, n). This integer piecewise linear programming problem in arbitrary dimension turns out to be equivalent to an integer non-linear (in fact, quadratic) optimization problem in dimension two. We also compute the sharp upper bound on a modification of the tropical determinant, where the maximum over all the transversals in a matrix is replaced with the minimum.</p>
Fri, 20 Feb 2015 08:00:00 +0000https://works.bepress.com/ivan-soprunov/11/ArticlesLattice Polytopes in Coding Theoryhttps://works.bepress.com/ivan-soprunov/19/<p>In this paper we discuss combinatorial questions about lattice polytopes motivated by recent results on minimum distance estimation for toric codes. We also prove a new inductive bound for the minimum distance of generalized toric codes. As an application, we give new formulas for the minimum distance of generalized toric codes for special lattice point configurations.</p>
Thu, 01 Jan 2015 08:00:00 +0000https://works.bepress.com/ivan-soprunov/19/Articles