Signed Minkowski decompositions of convex polygons into minimum simplices and factorization of max-plus functions

TL;DR

This paper explores signed Minkowski decompositions of convex polygons into minimal components and demonstrates how these decompositions relate to the factorization of max-plus functions, providing new insights into tropical geometry and function simplification.

Contribution

It proves that any 2D integral polygon admits a signed Minkowski decomposition into basic elements and shows how this leads to simplified representations of two-variable max-plus functions.

Findings

Any 2D integral polygon can be decomposed into integral points, segments, and triangles.

Max-plus functions with two variables and integer coefficients can be expressed in a simplified form.

The results connect Minkowski decompositions with tropical function factorizations.

Abstract

Signed Minkowski decomposition is an expression of a polytope as a Minkowski sum and difference of smaller polytopes. Signed Minkowski decompositions of a polytope can be interpreted as factorizations of a max-plus (tropical) function. We review two relations about Minkowski decompositions, and we prove that any 2-dimensional integral polytopes (polygons) have a signed Minkowski decomposition which consists of integral points, integral line segments of length 1, and integral triangles of area 1/2. From this result, we also obtain that any max-plus functions with two variables and integer coefficients can be expressed as a set of specified form of simpler max-plus functions.