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

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

2004-09-19

arxiv.org/abs/math/0409331v1

Mathematics

Number Theory

43 pages, 10 Figures

Scientific paper

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

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