Minimal primes of ideals arising from conditional independence statements
Let \(n, r_1, \dots, r_n\) be positive integers and set \(N=[r_1]\times \dots \times [r_n]\). Let \(R\) denote the polynomial ring over a field in variables \(x_a\) as \(a\) varies over \(N\). Let \(L\subseteq [n]\) and \(a, b \in N\). The switch function \(s(L,a,b)\) is an element of \(N\) whose \(i\)th component is \(s(L,a,b)_i=b_i\) if \(i\in L\) and \(s(L,a,b)_i=a_i\) otherwise. If \(L=\{j\}\), we simply write \(s(L,a,b)=s(j,a,b)\). For any two indices \(a, b\) in \(N\) we define the distance between them to be \[ d(a, b)=\# \{i: a_i\neq b_i\}. \] Also, for any \(L\subseteq [n]\) and \(i\in [n]\) we define: \[ f_{L, a, b}=x_ax_b-x_{s(L,a,b)}x_{s(L,b,a)}, \] \[ f_{i, a, b}=x_ax_b-x_{s(i,a,b)}x_{s(i,b,a)}. \] For any \(t\in [n]\), define \[ I^{\langle t \rangle}=\langle f_{i, a, b}: a, b \in N, d(a, b)=2, i\in [t]\rangle, \] \[ \widetilde{I^{\langle t \rangle}}=\langle f_{i, a, b}: a, b \in N, i\in [t]\rangle. \] A subset \(S\) of \(N\) is called \(t\)-switchable if for all \(a, b\in S\) with \(d(a, b)=2\), if \(i\in [t]\), then \(s(i,b,a)\in S\). Moreover, for a \(t\)-switchable subset \(S\) of \(N\), we define \[ \widetilde{I^{\langle t \rangle}}_S=\langle f_{i, a, b}: a, b\,\,\;\text{connected in}\, S \}, \,\,\, \mathrm{Var}^{\langle t \rangle}_S=\langle x_a: a\not\in S\rangle, \,\,\, \text{and}\,\,\, \widetilde{P^{\langle t \rangle}}_S=\mathrm{Var}^{\langle t \rangle}_S+\widetilde{I^{\langle t \rangle}}_S. \] Here two elements \(a, b\in S\) are connected in \(S\) if there is \[ a_0=a, a_1, \dots, a_{k-1}, a_k=b\in S, \] such that for all \(1\leq j\leq k\), \(a_{j-1}\) and \(a_j\) differ only in one component. We say that a \(t\)-switchable subset \(S\) of \(N\) is maximal if for all \(t\)-switchable subset \(T\) of \(N\) properly containing \(S\), \(P^{\langle t \rangle}_S\) and \(P^{\langle t \rangle}_T\) are incomparable. In this interesting paper, the authors by using a new combinatorial structure and \(t\)-switchable subsets, describe of the prime ideals minimal over \(I^{\langle t \rangle}\). More precisely, they prove that the set of prime ideals minimal over \(I^{\langle t \rangle}\) equals the set of ideals of the form \(P^{\langle t \rangle}_S\) as \(S\) varies over the maximal \(t\)-switchable subsets \(S\) of \(N\). Moreover, they prove that the minimal components of \(I^{\langle n \rangle}\) are prime ideals. Finally, the authors introduce a notion of diagonal orders for generic hypermatrices and they use these orders to give Gröbner basis for \(\widetilde{I^{\langle t \rangle}}\).
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