Analytic Representations in the 3-dim Frobenius Problem

Abstract

We consider the Diophantine problem of Frobenius for semigroup $\mathsf{S}({\mathbf{d}}^3)$ where ${\mathbf{d}}^3$ denotes the tuple $(d_1,d_2,d_3)$, $\gcd (d_1,d_2,d_3)=1$. Based on the Hadamard product of analytic functions we have found the analytic representation for the diagonal elements $a_{kk}({\mathbf{d}}^3)$ of the Johnson's matrix of minimal relations in terms of $d_1,d_2,d_3$. Bearing in mind the results of the recent paper this gives the analytic representation for the Frobenius number $F({\mathbf{d}}^3)$, genus $G({\mathbf{d}}^3)$ and the Hilbert series $H({\mathbf{d}}^3;z)$ for the semigroups $\mathsf{S}({\mathbf{d}}^3)$. This representation does complement the Curtis' theorem on the non-algebraic representation of the Frobenius number $F({\mathbf{d}}^3)$. We also give a procedure to calculate the diagonal and off-diagonal elements of the Johnson's matrix.

16 pages, 3 figures