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 f1x1+ ···+fnxn mod b ≤ g1x1+ ···+ gnxn, where g1,...,gn, f1,...,fn∈ Z, and b ∈ N. In this work, a geometrical characterization of these semigroups is given. On the basis of this geometrical approach, some algorithms are provided to check if a semigroup S in N n, with N^n \ S a finite set, is a proportionally modular affine semigroup