A geometrical characterization of proportionally modular affine semigroups

Abstract

A proportionally modular affine semigroup is the set of nonnegative integer solutions of a modular Diophantine inequality $f_1{x}_1+\cdots +f_n{x}_n\mathrm{mod}b\le g_1{x}_1+\cdots +g_n{x}_n$ where $g_1,\dots ,g_n,$ $f_1,\dots ,f_n\in \mathbb{Z}$ and $b\in \mathbb{N}$. In this work, a geometrical characterization of these semigroups is given. Moreover, some algorithms to check if a semigroup $S$ in ${\mathbb{N}}^n$, with ${\mathbb{N}}^n\setminus S$ a finite set, is a proportionally modular affine semigroup are provided by means of that geometrical approach.

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