# Lecture by Papa Sissokho (Illinois State University): Geometry of the minimal solutions of a linear Diophantine Equation

Let a_{1},...,a_{n} and b_{1},...,b_{m} be fixed positive integers, and let S denote the set of all nonnegative integer solutions of the equation x_{1}a_{1}+...+x_{n}a_{n}=y_{1}b_{1}+...+y_{m}b_{m}. A solution (x_{1},...,x_{n},y_{1},...,y_{m}) in S is called *minimal* if it cannot be expressed as the sum of two nonzero solutions in S. For each pair (i,j), with 1 ≤ i ≤ n and 1 ≤ j ≤ m, the solution whose only nonzero coordinates are x_{i} = b_{j} and y_{j} = a_{i} is called a *generator*. We show that every minimal solution is a convex combination of the generators and the zero-solution. This proves a conjecture of Henk-Weismantel and, independently, Hosten-Sturmfels.

### Time & Location

Jul 05, 2021 | 02:15 PM

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