Frobenius Problem for Semigroups ${\sl S}(d_1,d_2,d_3)$

Abstract

The matrix representation of the set $\Delta ({\mathbf{d}}^3)$, ${\mathbf{d}}^3=(d_1,d_2,d_3)$, of the integers which are unrepresentable by $d_1,d_2,d_3$ is found. The diagrammatic procedure of calculation of the generating function $\Phi ({\mathbf{d}}^3;z)$ for the set $\Delta ({\mathbf{d}}^3)$ is developed. The Frobenius number $F({\mathbf{d}}^3)$, genus $G({\mathbf{d}}^3)$ and Hilbert series $H({\mathbf{d}}^3;z)$ of a graded subring for non--symmetric and symmetric semigroups $\mathsf{S}({\mathbf{d}}^3)$ are found. The upper bound for the number of non--zero coefficients in the polynomial numerators of Hilbert series $H({\mathbf{d}}^m;z)$ of graded subrings for non--symmetric semigroups $\mathsf{S}({\mathbf{d}}^m)$ of dimension, $m\ge 4$, is established.

43 pages, 10 Figures