Frobenius Number Of Almost Symmetric Numerical Generalized Almost Arithmetic Semigroups

Abstract

Let a, k, h, c be positive integers and d a non zero integer. Recall that a numerical generalized almost arithmetic semigroup S is a semigroup minimally generated by relatively prime positive integers a, ha + d, ha + 2d, . . . , ha + kd, c, that is its embedding dimension is k + 2. In a previous work, the authors described the Ap{é}ry set and a Gr{ö}bner basis of the ideal defining S under one technical assumption, the complete version will be published in a forthcoming paper. In this paper we continue with this assumption and we describe the Pseudo Frobenius set. As a consequence we give a complete description of S when it is symmetric or almost symmetric as well as generalize and extend the previous results of Ignacio Garc{í}a-Marco, J. L. Ram{í}rez Alfons{í}n and O. J. R{ø}dseth; we also find a quadratic formula for its Frobenius number that generalizes some results of J.C. Rosales, and P.A. Garc{í}a-S{á}nchez. Moreover, for given numbers a, d, k, h, c, a simple algorithm allows us to determine if S is almost symmetric or not and furthermore to find its type and Frobenius number.