Geometry of the minimal solutions of a linear Diophantine Equation

TL;DR

This paper characterizes the structure of minimal solutions to a linear Diophantine equation, proving they are convex combinations of certain generator solutions and the zero solution, confirming a conjecture by Henk-Weismantel and Hoşten-Sturmfels.

Contribution

It proves that all minimal solutions are convex combinations of generators and the zero solution, resolving a conjecture in the geometry of Diophantine equations.

Findings

Every minimal solution is a convex combination of generators and zero.

The result confirms the conjecture of Henk-Weismantel and Hoşten-Sturmfels.

Provides a geometric understanding of minimal solutions in linear Diophantine equations.

Abstract

Let $a_1,\ldots,a_n$ and $b_1,\ldots,b_m$ be fixed positive integers, and let ${\mathcal S}$ denote the set of all nonnegative integer solutions of the equation $x_1a_1+\ldots +x_na_n=y_1b_1+\ldots +y_mb_m$. A solution $(x_1,\ldots,x_n,y_1,\ldots,y_m)$ in ${\mathcal S}$ is called $\textit{minimal}$ if it cannot be expressed as the sum of two nonzero solutions in ${\mathcal S}$. For each pair $(i,j)$ with $1\leq i\leq n$ and $1\leq j\leq m$, the solution whose only nonzero coordinates are $x_i=b_j$ and $y_j=a_i$ is called a $\textit{generator}$. Our main result shows that every minimal solution is a convex combination of the generators and the zero-solution. This proves a conjecture of Henk-Weismantel and, independently, Ho\c{s}ten-Sturmfels.