On systems of binomials in the ideal of a toric variety (Q2758968)

Let \(k\) be any field and \(D\) be an \(m\times n\) matrix with non-negative integer entries \(d_{ij}\) with non-zero columns. The toric set \(\Gamma\) determined by \(D\) is the set in the affine space of dimension \(n\) given parametrically by \(x_i=t_1^{d_{1i}}t_2^{d_{2i}}\cdots t_m^{d_{mi}}\) for all \(i\). The ideal of functions vanishing on \(\Gamma\) is denoted by \(P\), in the polynomial ring in the \(x_i\)'s, the toric ideal of \(k[\Gamma]\). The matrix \(D\) also determines a natural homomorphism \(\psi: {\mathbb Z}^n\to{\mathbb Z}^m\). Let \(K_{\Gamma}\) denote the kernel of \(\psi\). If \(g=x^{\alpha}-x^{\beta}\) is a binomial, where \(\alpha,\beta\) are vectors with non-negative entries, then \(g\in P\) if and only if \(\widehat{g}=\alpha-\beta\in K_{\Gamma}\). The following is the main result of this article under review. Let \(g_1,\ldots, g_r\) be a set of binomials in the toric ideal \(P\) and let \(I\) be the ideal generated by the \(g_i\)'s.NEWLINENEWLINENEWLINELet \(G\) the subgroup of \(K_{\Gamma}\) generated by \(\widehat{g}_i\)'s. If \(\text{char}(k)=p\neq 0\) (respectively \(\text{char}(k)=0\)), then \(\text{rad}(I)=P\) if and only if: NEWLINENEWLINENEWLINE(a) \(p^mK_{\Gamma}\subset G\) for some non-negative integer \(m\) (respectively \(K_{\Gamma}= G\)), NEWLINENEWLINENEWLINE(b) \(\text{rad}(I,x_i)=\text{rad}(P,x_i)\) for all \(i\).