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Report Saremi 2015

I started my project by studying about two combinatorial concepts: Hilbert Depth and Hilbert Regularity.

  In [1] Bruns et al. Introduced a  type of decomposition, which they call Hilbert  decomposition, since it only depends on the Hilbert function of M, and an analogous notion of depth, called Hilbert depth. In the other paper of Bruns et al. [2]; they introduced Hilbert regularity of M as the lowest possible value of the Castelmuovo-Mumford regularity for an R- module with Hilbert series H_M. Then they gave an algorithm for the computation of the Hilbert regularity and Hilbert depth of an R-module.
Firstly, during last two months, I studied many papers in this area and  then I learn computing the Hilbert depth and Hilbert Regularity  by Prof. Uliczka's help and then I was familiar with working by two computational mathematics systems: Macaylay 2 and Cocoa.

 Nowadays, I’m studying about 'combinatorial commutative algebra ', a fascinating new branch of commutative algebra created by Hochster and Stanley in the mid-seventies. The combinatorial objects considered are simplicial complexes to which one assigns algebraic objects, the Stanley-Reisner rings. The most important invariant of simplicial complexes, its f-vector, can be easily transformed into the h-vector, an invariant enclosed by the Hilbert function of the associated Stanley-Reisner ring.

 We study how the face numbers of simplicial complexes, f-vector, are related to the Hilbert regularity of the corresponding  Stanley-Reisner ring. 
For computing a Hilert regularity of Stanley-Reisner ring, we use the (n, k)-boundary presentations  of Hilbert series that introduced in [2]. 
We checked 1-dimensional simplicial complexes and 2-dimensional simplicial complexes.

  [1]. W. Bruns, C.Krattenthaler and J. Uliczka; Stanley decomposition and Hilbert depth in the Koszul complex.  J. Commut. Algebra 2, 327 – 35(2010).

 [2]. W. Bruns, J. J. Moyano-Fernandez and J. Uliczka,  Hilbert Regularity of Z-graded Modules Over Polynomial Rings. Accepted in J. Commut. Algebra