Toric fiber products
Let \(s_1, \ldots, s_r, t_1, \ldots, t_r\) be positive integers and consider two sets of variables \[ \{x_{ij}\}_{1\leq i\leq r, 1\leq j\leq s_i} = :x,\;\{y_{ij}\}_{1\leq i\leq r, 1\leq j\leq t_i} = :y. \] Define a multigraduation by \(\deg(x_{ij})=\deg (y_{ik})=a^{(i)}\in\mathbb Z^d\) and assume there exist \(\omega\in\mathbb Q^d\) such that \({}^t\omega\cdot a^{(i)}=1\) for all \(i\). Let \(\mathcal A=\{a^{(1)}, \ldots, a^{(r)}\}\) and \(I, J\) be homogeneous ideals in \(K[x]\) and \(K[y]\) respectively. Let \(R=K[x]/I\) and \(S=K[y]/J\) and \(\Phi_{I,J}:K[z]\to R\otimes_K S\) be defined by \(\Phi_{I,J}(z_{ijk})=x_{ij}\otimes y_{ik}\) with \(z=(z_{ijk})_{1\leq i\leq r, 1\leq j\leq s_i, 1\leq k\leq t_i}\). The kernel of \(\Phi_{I,J}\) is called the toric fibre product \(I\times_\mathcal A J\) of \(I\) and \(J\). A Gröbner basis of \(I\times_\mathcal A J\) is explicitly constructed in terms of a Gröbner basis of \(I\) and a Gröbner basis of \(J\) under the assumption that \(\mathcal A\) is a linearly independent set. The result is applied to algebraic statistics.
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