A “pseudo-polynomial” algorithm for the Frobenius number and Gröbner basis
Un Pseudo-polynomial algorithme pour calculer le Nombre de Frobenius et Bases de Groebner
Résumé
Given $n\geqslant 2$
and $a_1,\ldots,a_n\in \bn.$ Let $S=\left\langle a_1,\ldots,a_n\right\rangle $ be a semigroup. The aim of this paper is to give an effective pseudo-polynomial algorithm on $a_1$, which computes the Ap\'ery set and the Frobenius number of $S$. We also find the \gbb of the toric ideal defined by $S$, for the weighted degree reverse lexicographical order $\prec _{w}$ to $x_1,\ldots ,x_n$, without using Buchberger's algorithm. As an application we introduce and study some special classes of semigroups. Namely, when $S$ is generated by generalized arithmetic progressions and generalized almost arithmetic progressions with the ratio a positive or a negative number. We determine symmetric and almost symmetric semigroups generated by a generalized arithmetic progression.