A normal (respectively, graded normal) vector configuration A defines the toric ideal I A of a normal (respectively, projectively normal) toric variety. These ideals are Cohen-Macaulay, and when A is normal and graded, I A is generated in degree at most the dimension of I A . Based on this, Sturmfels asked if these properties extend to initial ideals-when A is normal, is there an initial ideal of I A that is Cohen-Macaulay, and when A is normal and graded, does I A have a Gröbner basis generated in degree at most dim(I A ) ? In this paper, we answer both questions positively for -normal configurations. These are normal configurations that admit a regular triangulation with the property that the subconfiguration in each cell of the triangulation is again normal. Such configurations properly contain among them all vector configurations that admit a regular unimodular triangulation. We construct non-trivial families of both -normal and non--normal configurations.
Bousquet-Mélou & Eriksson's lecture hall theorem generalizes Euler's celebrated distinctodd partition theorem. We present an elementary and transparent proof of a refined version of the lecture hall theorem using a simple bijection involving abacus diagrams.
If C is a clutter with n vertices and q edges whose clutter matrix has column vectors A = {v1, . . . , vq}, we call C an Ehrhart clutter if {(v1, 1), . . . , (vq, 1)} ⊂ {0, 1} n+1 is a Hilbert basis. Letting A(P ) be the Ehrhart ring of P = conv(A), we are able to show that if C is a uniform unmixed MFMC clutter, then C is an Ehrhart clutter and in this case we provide sharp upper bounds on the Castelnuovo-Mumford regularity and the a-invariant of A(P ). Motivated by the Conforti-Cornuéjols conjecture on packing problems, we conjecture that if C is both ideal and the clique clutter of a perfect graph, then C has the MFMC property. We prove this conjecture for Meyniel graphs, by showing that the clique clutters of Meyniel graphs are Ehrhart clutters. In much the same spirit, we provide a simple proof of our conjecture when C is a uniform clique clutter of a perfect graph. We close with a generalization of Ehrhart clutters as it relates to total dual integrality.
In this paper we provide new characterizing properties of TDI systems. A corollary is Sturmfels' theorem relating toric initial ideals generated by square-free monomials to unimodular triangulations. A reformulation of these test-sets to polynomial ideals actually generalizes the existence of square-free monomials to arbitrary TDI systems, providing new relations between integer programming and Gröbner bases of toric ideals. We finally show that stable set polytopes of perfect graphs are characterized by a refined fan that is a triangulation consisting only of unimodular cones, a fact that endows the Weak Perfect Graph Theorem with a computationally advantageous geometric feature. Three ways of implementing the results are described and some experience about one of these is reported.
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