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Thursday, June 13, 2019

**Abstract:** Let S be a numerical semigroup. Its embedding dimension e(S) is the minimal number of generators, the Frobenius number F(S) is the largest integer not in S , and n(S) counts the elements in S that are < F(S). Wilf's conjecture states that F(S) < e(S)n(S). It has been proved in many cases, but remains a major open problem in the combinatorial theory of numerical semigroups. We will show that for fixed multiplicity m=m(S), the smallest nonzero element of S, the conjecture can be decided algorithmically by polyhedral methods using the parametrization of multiplicity m semigroups by the lattice points of the Kunz polyhedron P(m). With them we have verified the conjecture for m up to 18.