On $p$-Frobenius of affine semigroups

The aim of this paper is to study the $p$-Frobenius vector of affine semigroups $S\subset {\mathbb{N}}^q$; that is, the maximum element, with respect to a graded monomial order, with at most $p$ factorizations in $S$. We produce several algorithms to compute these vectors. Finally, we study how the $p$-Frobenius vectors behave when considering gluings of $S$ with ${\mathbb{N}}^q$.