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seminars:comb:abstract.200202beck

Given relatively prime positive integers a_{1} , …, a_{n} , we call an integer t *representable* if there exist nonnegative integers m_{1} , …, m_{n} such that

t = m_{1} a_{1} + … + m_{n} a_{n} .

We study the *linear diophantine problem of Frobenius*: namely, to find the largest integer which is not representable.

We translate this problem into a geometric one: consider N(t), the number of nonnegative integer solutions (m_{1} , …, m_{n} ) to m_{1} a_{1} + … + m_{n} a_{n} = t for any positive integer t. N(t) enumerates the integer points in a polytope. Solving the Frobenius problem now simply means finding the largest zero of N(t). N(t) turns out to be a quasi-polynomial, thereby yielding a straightforward analytic tool to recover and extend some well-known results on this classical problem.

seminars/comb/abstract.200202beck.txt · Last modified: 2020/01/29 14:03 (external edit)

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