A Geometrical Characterization of Proportionally Modular Affine Semigroups

摘要

A proportionally modular affine semigroup is the set of nonnegative integer solutions of a modular Diophantine inequality f(1)x(1) + ... + f(n)x(n) mod b <= g(1)x(1) + ... + g(n)x(n), where g(1),..., g(n), f(1),..., f(n) is an element of Z, and b is an element of 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-nS a finite set, is a proportionally modular affine semigroup.