## Description

A numerical semigroup S is an additive subsemigroup of the positive integers. The family of shifted numerical semigroups S(n) is found by adding a positive integer n to each generator of S. Each numerical semigroup is associated with a homogeneous toric ideal, constructed from the minimal relations of its generators, while every toric ideal associates with a finite set of reduced Gr ̈obner basis. The Gr ̈obner fan of an ideal is the geometric representation of the set of all reduced Gr ̈obner bases of the ideal. In this thesis, we algorithmically characterize Gr ̈obner fans of toric ideals of 3-generator shifted numerical semigroups when n is suciently large. We then use this algorithm to prove the number of facets of the Gr ̈obner fan and the number of generators in the universal Gr ̈obner basis are quasi-linear as functions of n.