Gröbner bases of contraction ideals
For any field \(K\) and any \(m \times n\) integer matrix \(\mathcal{A}=(\mathbf{a}_1,\ldots,\mathbf{a}_n)\), where \(\mathbf{a}_j = {}^t(a_{1j},\ldots,a_{mj})\), the author considers the \(K\)-algebra homomorphism \(\phi_{\mathcal{A}}:K[x_1,\ldots,x_n] \rightarrow K[y_1^{\pm 1},\ldots, y_m^{\pm 1}]\) sending \(x_j\) to \(\mathbf{y}^{\mathbf{a}_j}\). The ideal \(P_{\mathcal{A}} := \text{Ker}\, \phi_{\mathcal{A}}\) is called \textit{toric ideal} of \(\mathcal{A}\). The paper investigates toric ideals of the product of two matrices \(\mathcal{C}=\mathcal{B}\cdot\mathcal{A}\). As \(\phi_{\mathcal{C}} = \phi_{\mathcal{B}\cdot\mathcal{A}} = \phi_{\mathcal{B}}\circ\phi_{\mathcal{A}}\), we have that \(P_{\mathcal{C}} = P_{\mathcal{B}\cdot\mathcal{A}} = \phi_{\mathcal{A}}^{-1}(P_{\mathcal{B}})\), hence the author studies \textit{contraction ideals} under the monomial homomorphism \(\phi_{\mathcal{A}}\) and in particular their initial ideals. Consider a homomorphism \(\phi_{\mathcal A}\), associated to an integer matrix \(\mathcal{A}\) with non-negative entries, and a weight vector \(\mathbf{w} = (w_1,\ldots,w_m)\) giving a \(\mathbb{N}\)-graded structure on \(K[y_1,\ldots,y_m]\). The vector \(\phi_{\mathcal{A}}^{\ast}\mathbf{w} := \mathbf{w}\cdot\mathcal{A} = (\deg_{\mathbf w}\phi_{\mathcal{A}}(x_1),\ldots,\deg_{\mathbf{w}}\phi_{\mathcal{A}}(x_m))\) induces a \(\mathbb{N}\)-graded structure on \(K[x_1,\ldots,x_m]\). Firstly, in order to reduce to the case of monomial ideals, the author discusses under which conditions the computation of the initial ideal commutes with the contraction, i.e. when \[ \text{in}_{\phi_{\mathcal{A}}^{\ast}\mathbf{w}}\big( \phi_{\mathcal{A}}^{-1}(I)\big) = \phi_{\mathcal{A}}^{-1}\big(\text{in}_{\mathbf{w}}(I)\big). \] Then, the main result of the paper concerns \(\mathbb{Z}^d\)-graded polynomial rings \(K[y_1,\ldots,y_m]\), with \(\deg_{\mathbb{Z}^d}(y_i) = \mathbf{v}_i \in \mathbb{Z}^d\). For any finitely generated subsemigroup \(\mathfrak{H} \subset \mathbb{Z}^d\), the author considers matrices \(\mathcal{A}_{\mathfrak{H}}\) whose columns represent a system of generators of the semigroup \(\{\mathbf{a} \in \mathbb{N}^m \;|\;\deg_{\mathbb{Z}^d}(\mathbf{y}^{\mathbf{a}}) \in \mathfrak{H}\}\). In this setting, for any \(\mathbb{Z}^d\)-graded ideal \(I \subset K[y_1,\ldots,y_m]\), it is proved that: the initial ideal of the contraction ideal \(\phi^{-1}_{\mathcal{A}_{\mathfrak{H}}}(I)\) is generated in degree at most \(m\), if both \(I\) and \(P_{\mathcal{A}_{\mathfrak{H}}}\) have this property, and the initial ideal of \(\phi^{-1}_{\mathcal{A}_{\mathfrak{H}}}(I)\) is square-free, if so are the initial ideals of \(I\) and \(P_{\mathcal{A}_{\mathfrak{H}}}\).
scientific article; zbMATH DE number 6062490
scientific article