On the number of equations defining certain varieties (Q1353048)

The author considers the minimal number of equations needed to define the variety \(V(I)\) associated to a monomial ideal \(I\) of a polynomial ring, this number is called the arithmetical rank and denoted by \(\text{ara}(I)\). The author generalizes a lemma of \textit{Th. Schmitt} and \textit{W. Vogel} [Math. Ann. 245, 247-253 (1979; Zbl 0408.14014)]; given conditions on a decomposition of a system of generators of \(I\), she can construct a system \(J\) of elements in \(I\) with cardinal less than the minimal number of generators of \(I\) and such that \(\text{rad}(J)= \text{rad}(I)\), where \((J)\) is the ideal generated by \(J\). This criterium has several interesting applications to a class of ideals, where the author computes \(\text{ara}(I)\).