No abstract

McMullen's proof of the Hard Lefschetz Theorem for simple polytopes is studied, and a new proof of this theorem that uses conewise polynomial functions on a simplicial fan is provided.

International audience The present paper connects sharpenings of Sauer's bound on forbidden configurations with color critical hypergraphs. We define a matrix to be \emphsimple if it is a $(0,1)-matrix$ with no repeated columns. Let $F$ be $a k× l (0,1)-matrix$ (the forbidden configuration). Assume $A$ is an $m× n$ simple matrix which has no submatrix which is a row and column permutation of $F$. We define $forb(m,F)$ as the best possible upper bound on n, for such a matrix $A$, which depends on m and $F$. It is known that $forb(m,F)=O(m^k)$ for any $F$, and Sauer's bond states that $forb(m,F)=O(m^k-1)$ fore simple $F$. We give sufficient condition for non-simple $F$ to have the same bound using linear algebra methods to prove a generalization of a result of Lovász on color critical hypergraphs.

We say a matrix is simple if it is a (0,1)-matrix with no repeated columns. Given m and a k×l (0,1)-matrix F we define forb(m, F ) as the maximum number of columns in a simple m-rowed matrix A for which no k × l submatrix of A is a row and column permutation of F . In set theory notation, F is a forbidden trace. For all k-rowed F (simple or nonsimple) Füredi has shown that forb(m, F ) is O(m k ). We are able to determine for which k-rowed F we have that forb(m, F ) is O(m k−1 ) and for which k-rowed F we have that forb(m, F ) is Θ(m k ).We need a bound for a particular choice of F . Define D12 to be the k ×(2 k −2 k−2 −1) (0,1)-matrix consisting of all nonzero columns on k rows that do not haveˆ1 1˜i n rows 1 and 2. Let 0 denote the column of k 0's. Define F k (t) to be the concatenation of 0 with t + 1 copies of D12. We are able to show that forb(m, F k (t)) is Θ(m k−1 ). Linear algebra methods and indicator polynomials originated in this context in a paper of the authors and Füredi and Sali. We provide a novel application of these methods.The results are further evidence for the conjecture of Anstee and Sali on the asymptotics for fixed F of forb(m, F ).