Comments on: Cohen Macaulay Algebras and Non-negativity http://benbraun.wordpress.com/2008/07/18/cohen-macaulay-algebras-and-non-negativity/ Sun, 28 Dec 2008 12:32:12 +0000 http://wordpress.com/ hourly 1 By: Lovasz’s Two Families Theorem « Combinatorics and more http://benbraun.wordpress.com/2008/07/18/cohen-macaulay-algebras-and-non-negativity/#comment-6 Lovasz’s Two Families Theorem « Combinatorics and more Sun, 28 Dec 2008 12:32:12 +0000 http://benbraun.wordpress.com/?p=12#comment-6 [...] 1975 theorem which applies for a larger class of simplicial complexes called “Cohen-Macaulay complexes”.   Possibly related posts: (automatically generated)Extremal Combinatorics IV: [...] [...] 1975 theorem which applies for a larger class of simplicial complexes called “Cohen-Macaulay complexes”.   Possibly related posts: (automatically generated)Extremal Combinatorics IV: [...]

]]> By: Gil Kalai http://benbraun.wordpress.com/2008/07/18/cohen-macaulay-algebras-and-non-negativity/#comment-2 Gil Kalai Sat, 19 Jul 2008 18:14:26 +0000 http://benbraun.wordpress.com/?p=12#comment-2 This is a very nice post! As an appetizer for more stuff that Ben may discuss later on his blog and I plan to discuss on my blog, let me mention the following. Consider a simple d-dimensional polytope P and a linear objective function w which is not constant on any edge of P. Every vertex v of P is connected by an edge of the polyope to d other vertices and let deg(v) be the number of those vertices u adjecent to v for which has w(u)>w(v). Let $latex h_k(P)$ be the number of vertices v so that deg(v)=k. The polynomial $latex h_0(P)+h_1(P)x+ H_2(P)x^2+\dots +,h_d(P)x^d$ is closely related to the polynomial in the last line of the post! This is a very nice post! As an appetizer for more stuff that Ben may discuss later on his blog and I plan to discuss on my blog, let me mention the following. Consider a simple d-dimensional polytope P and a linear objective function w which is not constant on any edge of P. Every vertex v of P is connected by an edge of the polyope to d other vertices and let deg(v) be the number of those vertices u adjecent to v for which has w(u)>w(v). Let h_k(P) be the number of vertices v so that deg(v)=k.

The polynomial h_0(P)+h_1(P)x+ H_2(P)x^2+\dots +,h_d(P)x^d is closely related to the polynomial in the last line of the post!

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