A note on the computation of the Frobenius number of a numerical semigroup. (Q1941749)

Let \(n_1,\dots,n_d\) be positive integers with greatest common divisor one, and let \(\Gamma\) be the numerical semigroup spanned by them, that is \(\Gamma=\mathbb Nn_1+\cdots+\mathbb Nn_d\), with \(\mathbb N\) the set of nonnegative integers. Since \(\gcd(n_1,\dots,n_d)=1\), it can be shown that \(\mathbb N\setminus\Gamma\) has finitely many elements. Thus \(\max(\mathbb Z\setminus\Gamma)\) exists (where \(\mathbb Z\) denotes the set of integers), and this maximum is known as the Frobenius number of \(\Gamma\), denoted in this manuscript by \(f(\Gamma)\). For a nonzero element \(n\in\Gamma\), the Apéry set of \(n\) in \(\Gamma\) is defined as \(\mathrm{Ap}(\Gamma,n)=\{h\in\Gamma\mid h-n\not\in\Gamma\}\). This set has finitely many elements, and it is well known that \(f(\Gamma)=\max\mathrm{Ap}(\Gamma,n)-n\). Let \(k\) be a field, \(R=k[X_1,\dots,X_d]\), and \(\pi\colon R\to k[t]\) be the graded homomorphism induced by \(X_i\mapsto t^{n_i}\), with \(\deg(X_i)=n_i\), \(i\in\{1,\dots,d\}\) and \(\deg(t)=1\). The image of \(\pi\) is denoted by \(k[\Gamma]\) which is referred as the semigroup ring associated to \(\Gamma\). The prime ideal \(\mathfrak p=\ker\pi\) is the presentation ideal of \(k[\Gamma]\), sometimes called the ideal associated with \(\Gamma\). The set \(M=\Gamma\setminus\{0\}\) is a \(\Gamma\)-ideal, in the sense that \(M+\Gamma\subseteq M\neq\emptyset\), indeed it is the unique maximal ideal of \(\Gamma\). Define \(M^-=\{z\in\mathbb Z\mid z+M\subseteq\Gamma\}\). Then \(M^-\) is also a \(\Gamma\)-ideal (containing properly \(\Gamma\)). If \(\Gamma\neq\mathbb N\), then \(M^-\subseteq\mathbb N\). The cardinality of the set \(M^-\setminus\Gamma\) is denoted by \(r(\Gamma)\). In the literature, this amount is also known as the type of \(\Gamma\), since it corresponds with the Cohen-Macaulay type of the semigroup ring \(k[\Gamma]\). Moreover, the elements in \(M^-\setminus\Gamma\) are usually called pseudo-Frobenius numbers of \(\Gamma\), since by definition for \(g\in M^-\setminus\Gamma\), \(g+(\Gamma\setminus\{0\})\subseteq\Gamma\) (it can be shown that \(M^-\setminus\Gamma=\mathrm{Maximals}_{\leq_\Gamma}(\mathrm{Ap}(\Gamma,n)-n)\), where \(h\leq_\Gamma h'\) if \(h'-h\in\Gamma\)). Denote by \(\mathfrak p'\) the image of \(\mathfrak p\) by the epimorphism from \(R\) onto \(k[X_1,\dots,X_{d-1}]\) obtained by mapping \(X_d\) to \(0\). Set \(R'=K[\Gamma]/(t^{n_d})\cong k[X_1,\dots,X_{d-1}]/\mathfrak p'\). The trivial submodule of \(R'\) is defined as \(\mathrm{Triv}(R')=\{x\in R'\mid x\cdot\mathfrak m_{R'}=(0)\}\), where \(\mathfrak m_{R'}\) is the unique maximal graded ideal of \(R'\). The author observes that \(r(\Gamma)=\dim_k\mathrm{Triv}(R')\), and thus there is a basis \(\mathfrak B=\{b_1,\dots,b_{r(\Gamma)}\}\) of \(\mathrm{Triv}(R')\). Let \(\beta\in\mathfrak B\) be such that \(\deg(\beta)=\max\{\deg(b_1),\dots,\deg(b_{r(\Gamma)})\}\). Then the author highlights that \(\deg(\beta)\) does not depend on the choice of \(\mathfrak B\), and gives the formula \(f(\Gamma)=\deg(\beta)-n_d\). As a corollary, \(\max\mathrm{Ap}(\Gamma,n_d)=\deg(\beta)\).