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Frobenius vectors, Hilbert series and gluings of affine semigroups. (English) Zbl 1350.20043

This paper deals with submonoids \(S=\langle\mathbf a_1,\ldots,\mathbf a_k\rangle\) of \((\mathbb N^d,+)\). In this review \(\{\mathbf a_1,\ldots,\mathbf a_k\}\) will always denote the unique minimal generating system of \(S\).
Let \(G(S)\) denote the subgroup of \((\mathbb Z^d,+)\) generated by \(S\). \(S\) is said to be a gluing of \(S_1\) and \(S_2\) by \(\mathbf d\) if there is a partition \(\{\mathbf a_1,\ldots,\mathbf a_k\}=A_1\sqcup A_2\) such that \(S_1=\langle A_1\rangle\), \(S_2=\langle A_2\rangle\), \(\mathbf d\in S_1\cap S_2\) and \(G(S_1)\cap G(S_2)=\mathbf d\mathbb Z\). Invariants of \(S\) are related to those of \(S_1\) and \(S_2\), in particular Frobenius vectors and Hilbert series.
An element \(\mathbf f\in\mathbb Z^d\) is called a Frobenius vector of \(S\) if \(\mathbf f+(G(S)\cap\{\sum_{i=1}^k\lambda_i\mathbf a_i:\lambda_i\in\mathbb Q^+\})\subseteq S\setminus\{\mathbf 0\}\). It is shown that if \(S\) is the gluing of \(S_1\) and \(S_2\) by \(\mathbf d\) and \(\mathbf f_1,\mathbf f_2\) are Frobenius vectors of \(S_1,S_2\), respectively, then \(\mathbf f_1+\mathbf f_2+\mathbf d\) is a Frobenius vector of \(S\).
It is shown that the (multigraded) Hilbert series \(H_S(\mathbf x)=\sum_{\mathbf s\in S}\mathbf x^{\mathbf s}\) of the toric ring \(\Bbbk[S]\subseteq\Bbbk[x_1,\ldots,x_d]\) is equal to \((1-\mathbf x^{\mathbf d})H_{S_1}(\mathbf x)H_{S_2}(\mathbf x)\) if \(S\) is the gluing of \(S_1\) and \(S_2\) by \(\mathbf d\).

MSC:

20M14 Commutative semigroups
11D07 The Frobenius problem
20M05 Free semigroups, generators and relations, word problems
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References:

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