Set-theoretic complete intersections on binomials (Q2782615)

Let \(k\) be a field, \(V\) an affine toric variety of codimension \(r\) over \(k\). Let also \(I\) be a binomial ideal. The binomial arithmetical rank of \(I\), \(\text{bar}(I)\) is the smallest integer \(s\), such that \(\sqrt{I}= \sqrt{(f_1,\dots, f_s)}\). Then \(\text{ht}(I)\leq \text{ara}(I)\leq\text{bar}(I)\) and \(I\) is called a set-theoretic complete intersection on binomials if \(\text{ht}(I)= \text{bar}(I)\). The paper deals with the problem of characterizing the varieties which are set-theoretic complete intersections on binomials. This is done separately in the zero and positive characteristic cases. Several examples are given.