Analytic representations in the three-dimensional Frobenius problem

Abstract

We consider the Diophantine problem of Frobenius for the semigroup $\mathsf{S}(\mathbf{d}^{3})$ , where d ³ denotes the triple (d ₁,d ₂,d ₃), gcd (d ₁,d ₂,d ₃)=1. Based on the Hadamard product of analytic functions, we find the analytic representation of the diagonal elements a _kk (d ³) of Johnson’s matrix of minimal relations in terms of d ₁, d ₂, and d ₃. With our recent results, this gives the analytic representation of the Frobenius number F(d ³), genus G(d ³), and Hilbert series H(d ³;z) for the semigroups $\mathsf{S}(\mathbf{d}^{3})$ . This representation complements Curtis’s theorem on the nonalgebraic representation of the Frobenius number F(d ³). We also give a procedure for calculating the diagonal and off-diagonal elements of Johnson’s matrix.