Search

Workshop

The volume polynomial of lattice polygons

  • Jenya Soprunova (Kent State University, USA)
E1 05 (Leibniz-Saal)

Abstract

Let P and Q be lattice polygons in the plane. Then their normalized volumes V(P) and V(Q) and their normalized mixed volume V(P,Q) satisfy the classical Minkowski inequality V(P)V(Q)<= V(P,Q)^2. We show that for any triple of nonnegative integers a,b,c that satisfy the inequality ab<= c^2 there exist lattice polygons P and Q in R^2 such that V(P)=a, V(Q)=b, and V(P,Q)=c. This shows that every indefinite quadratic form with non-negative integer coefficients is the volume polynomial of a pair of lattice polygons, which solves the discrete version of the Heine–Shephard problem for two bodies in the plane. As an application, we show how to construct a pair of planar tropical curves (or a pair of divisors on a toric surface) with given intersection number and self-intersection numbers.

This is joint work with Ivan Soprunov.

conference
29.07.24 02.08.24

MEGA 2024

MPI für Mathematik in den Naturwissenschaften Leipzig (Leipzig) E1 05 (Leibniz-Saal)
Universität Leipzig (Leipzig) Felix-Klein-Hörsaal

Mirke Olschewski

Max Planck Institute for Mathematics in the Sciences Contact via Mail

Saskia Gutzschebauch

Max Planck Institute for Mathematics in the Sciences Contact via Mail

Christian Lehn

Ruhr-Universität Bochum

Irem Portakal

Max Planck Institute for Mathematics in the Sciences

Rainer Sinn

Universität Leipzig

Bernd Sturmfels

Max Planck Institute for Mathematics in the Sciences

Simon Telen

Max Planck Institute for Mathematics in the Sciences

Upcoming Events of this Conference