Lecture by Papa Sissokho (Illinois State University): Geometry of the minimal solutions of a linear Diophantine Equation
Let a1,...,an and b1,...,bm be fixed positive integers, and let S denote the set of all nonnegative integer solutions of the equation x1a1+...+xnan=y1b1+...+ymbm. A solution (x1,...,xn,y1,...,ym) 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 xi = bj and yj = ai 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