A criterion for the primeness of ideals generated by polynomials with separated variables
Let \(K\) be an algebraically closed field of characteristic \(p\geq 0\), \(f_i=\sum^n_{j=1} f_{ij}(X_j)\in K[X_1,...,X_n]\), \(i=1,\dots,m\), polynomials with separated variables and \(V\) the closed subscheme of \(\mathbb{A}^n\) defined by \(f_i=0\), \(i=1,\dots,m\). Let \(\mu_j=\max \{\deg(f_{1j}), \dots,\deg (f_{mj})\}\), \(j=1,\dots,n\) and assume that \(p\) does not divide \(\mu_j\) if \(p>0\) and \(\mu_j\neq 0\) if \(p=0\). Consider the matrix \(A=(a_{ij})\), where \(f_{ij}= a_{ij} X_j^{\mu_j}+\cdots\). In the paper under review, the author shows that \(V\) is a variety (i.e., an irreducible and reduced closed subscheme of \(\mathbb{A}^n)\), if each non-trivial linear combination of the rows of \(A\) has at least three non-zero entries.
scientific article; zbMATH DE number 1389700