Frobenius number and minimal presentation of certain numerical semigroups

Abstract

Suppose e≥4 be an integer, a=e+1, b>a+(e−3)d, gcd(a,d)=1 and d∤(b−a). Let M={a,a+d,a+2d,…,a+(e−3)d,b,b+d}, which forms a minimal generating set for the numerical semigroup Γe(M), generated by the set M. We calculate the Ap\'{e}ry set and the Frobenius number of Γe(M). We also show that the minimal number of generators for the defining ideal p of the affine monomial curve parametrized by X0=ta, X1=ta+d,…,Xe−3=ta+(e−3)d, Xe−2=tb, Xe−1=tb+d is a bounded function of e.Ranjana Mehta, Joydip Saha, and Indranath Sengupt