The Frobenius vector of a free affine semigroup. (Q2909795)

An affine semigroup is a finitely generated submonoid of \(\mathbb N^e\) for some positive integer \(e\). An affine semigroup \(\Gamma\subseteq\mathbb N^e\) is simplicial if there is a generating system of the form \(\{v_1,\dots,v_e,v_{e+1},\dots,v_{e+s}\}\) such that the \(\mathbb R\)-vector space spanned by \(\{v_1,\dots,v_e\}\) is \(\mathbb R^e\), and \(\Gamma\subseteq\mathrm{cone}(\{v_1,\dots,v_e\})\), where \(\mathrm{cone}(\{v_1,\dots,v_e\})\) stands for the set of linear combinations of \(\{v_1,\dots,v_e\}\) with nonnegative real coefficients.NEWLINENEWLINE For \(k\in\{1,\dots,s\}\), let \(\Gamma_k\) be the submonoid of \(\Gamma\) generated by \(\{v_1,\dots,v_e,v_{e+1},\dots,v_{e+k}\}\), \(G_k\) be the group spanned by \(\Gamma_k\), and \(D_k\) be the greatest common divisor of the \((e,e)\) minors of the matrix whose rows are \(v_1,\dots,v_e,v_{e+1},\dots,v_{e+k-1}\).NEWLINENEWLINE The monoid \(\Gamma\) is free if and only if either \(k=0\) or \(\Gamma\) is the gluing of the free semigroup \(\Gamma_{s-1}\) and \(\langle v_{e+s}\rangle\) (the monoid generated by \(\{v_{e+s}\}\)), that is, \(\Gamma_{s-1}\) is free and there exists \(d\in\mathbb N^e\setminus\{0\}\) such that \(G_{s-1}\cap v_{e+s}\mathbb Z=d\mathbb Z\) and \(d\in\Gamma_{s-1}\cap\langle v_{e+s}\rangle\).NEWLINENEWLINE The author proves that \(\Gamma\) is free if and only if \(D_1>D_2>\cdots>D_{s+1}\) and for all \(k\in\{1,\dots,s\}\), \(\frac{D_k}{D_{k-1}}v_{e+k}\in\Gamma_{k-1}\).NEWLINENEWLINE A Frobenius vector of \(\Gamma\) is an element \(g\) in \(\mathbb Z^e\) such that \(g\not\in\Gamma\) and \(g+C_e\cap G_s\subseteq\Gamma\), where \(C_e\) is the interior of \(\mathrm{cone}(\{v_1,\dots,v_e\})\).NEWLINENEWLINE In the manuscript under review it is shown that a unique Frobenius vector exists for free simplicial affine semigroups, and a formula is given in terms of the Apéry sets of the extremal rays. This formula generalizes Selmer's formula for numerical semigroups.NEWLINENEWLINE The paper offers a series of applications and several examples that are useful to motivate and better understand the results presented.