The Hilbert Series of Algebras of the Veronese Type

Abstract

This paper gives an explicit formula for the Hilbert series of algebras of Veronese type.

Ehrhart Polynomials of Matroid Polytopes and Polymatroids

“…In the following we derive some new and useful equalities and prove symmetry and unimodality for the Katzman coefficients. Katzman [25] gave an explicit equation for the h * -vector of uniform matroid polytopes and the coefficients of their Ehrhart polynomials, although he did not use the same language. We restate it here for our purposes.…”

“…Our experiments included, among others, many examples coming from small graphical matroids, random realizable matroids over fields of small positive characteristic, and the classical examples listed in the Appendix of [32] for which we list To present our results about the h * -vector we begin explaining the details with the following numbers introduced in [25], which we refer to as the Katzman coefficients.…”

“…By far the most comprehensive study we made was for the family of uniform matroids. In this case we based our computations on the theory of Veronese algebras as developed by M. Katzman in [25]. There, Katzman gave an explicit equation for the h * -vector of uniform matroid polytopes (again, using the language of Veronese algebras).…”

“…See Corollary 2.9 in[25]) Let P(U r,n ) be the matroid polytope of the uniform matroid of rank r on n elements. Then the h * -polynomial of P(U r,n ) is given by That is, for h * (P(U n,r) )) = (h * 0 , .…”

“…We include this information here just for the 28 famous matroids presented in [32]. Results in [25], with some additional careful calculations, imply that Conjecture 2 is true for all rank 2 uniform matroids. Regarding part (A) of the conjecture, we were also able to prove partial unimodality for uniform matroids of rank 3.…”