Convex polytopes all of whose reverse lexicographic initial ideals are squarefree (Q2716095)

An integral convex polytope \(P\subset{\mathbb R}^n\) is called compressed if all its ``pulling triangulations'' are unimodular. Since all vertices of \(P\) belong to \({\mathbb Z}^n\) it is possible to associate to \(P\) a canonical toric ideal \(I_P\), generated by suitable binomials. The property of \(P\) of being compressed can be restated in terms of toric ideals: NEWLINENEWLINENEWLINE\(P\) is compressed if and only if the initial ideal of \(I_P\) with respect to any reverse lexicographic monomial order is generated by squarefree monomials. NEWLINENEWLINENEWLINEThe authors show the following result (theorem 1.1): Let \(a_{i,j}\), \(b_i\) and \(\varepsilon_i\), \(1\leq i\leq m\), \(1\leq j\leq n\), be integers with \(\varepsilon_i\in\{0,1\}\). Suppose that the convex polytope \(P\subset{\mathbb R}^n\) is defined by the system of linear inequalities \(b_i\leq\sum_{1\leq j\leq n}a_{ij}X_j\leq b_i+\varepsilon_i\), \(1\leq i\leq m\) and \(0\leq X_j\leq 1\), \(1\leq j\leq n\). Suppose furthermore that \(P\) is a (0,1)-polytope (i.e. its vertices belong to the hypercube \(\{0,1\}^n\)). Then \(P\) is compressed.NEWLINENEWLINENEWLINEIn case that the determinants of all square submatrices of the \((m\times n)\)-matrix \((a_{ij})_{1\leq i\leq m,1\leq j\leq n}\) belong to \(\{0,1,-1\}\), one concludes that \(P\) is a (0,1)-polytope and hence compressed (corollary 1.2). The proof of theorem 1.1 uses the characterization of compressed integral convex (0,1)-polytopes in terms of toric ideals.