Arithmetical rank of binomial ideals (Q2411675)

Let \(J\subseteq \mathbb{K}[x_1,\ldots,x_n]\) be a polynomial ideal. We remind that the arithmetical rank of \(J\), which is denoted by \(\text{ara}(J)\), is the smallest integer \(s\) for which there exist polynomials \(F_1,\ldots,F_s \in J\) such that \(\mathrm{rad}(J)=\mathrm{rad}(F_1,\ldots,F_s)\). Correspondingly, the binomial arithmetical rank which is denoted by \(\mathrm{bar}(J)\) is defined, where \(J\) is a binomial ideal of \(\mathbb{K}[x_1,\ldots,x_n]\). In this paper, the author studies the above notions for binomial ideals. Firstly, he gives lower bounds for \(\mathrm{ara}(J)\) and \(\mathrm{bar}(J)\) of a binomial ideal \(J\), in terms of a simplicial complex \(\Delta_J\) which he defines. In addition, he studies these notions in the special case of graphs ideals. The author proves that \(\mathrm{ara}(J_G)\geq n+l-2\), where \(G\) is a connected graph on the vertex set \([n]\) and \(l\) is its vertex connectivity. Also, he computes \(\mathrm{ara}(J_G)\) and \(\mathrm{bar}(J_G)\) for special cases of graphs.