Lattice points in Minkowski sums

TL;DR

This paper provides a shorter combinatorial proof for a lattice point decomposition in Minkowski sums of polygons, removing the previous smoothness condition on one of the polygons.

Contribution

It offers a simplified combinatorial proof of Fakhruddin's result, extending its applicability without the smoothness assumption.

Findings

Proof does not require P to be smooth

Lattice points in Minkowski sums can be decomposed without smoothness

Simplifies previous proofs of Fakhruddin's theorem

Abstract

Fakhruddin has proved that for two lattice polygons P and Q any lattice point in their Minkowski sum can be written as a sum of a lattice point in P and one in Q, provided P is smooth and the normal fan of P is a subdivision of the normal fan of Q. We give a shorter combinatorial proof of this fact that does not need the smoothness assumption on P.