Algorithmic and theoretical methods for studying monomial and binomial ideals
with applications in combinatorics, commutative algebra and graph theory
Project code PN-II-ID-PCE-2011-3-1023, contract nr.247/24.10.2011
A brief description of the research project
The proposed project will investigate questions in commutative algebra for which combinatorial and computational approaches are fruitful. In this context, we propose to study a variety of open problems related to the interplay between combinatorial and algebraic features of monomial and binomial ideals.
A primary goal of the proposed work is to identify broader classes of monomial ideals for which Stanley's conjecture holds, targetting the general case of all squarefree monomial ideals. Thanks to previous techniques developed by some of the team's members, we already have some partial results concerning the extension of some results obtained by Dorin Popescu from the case of squarefree monomial ideals to the larger class of all monomial ideals. Extensive computations that we made using the computer algebra softwares CoCoA and Singular indicate that one should expect that the sdepth of the polarization of an arbitrary monomial ideal is precisely the sdepth of the monomial ideal plus the number of the new added variables. This would imply at its turn that it is enough to prove Stanley's conjecture for squarefree monomial ideals in order to have it validated for arbitrary monomial ideals. Recently, Dorin Popescu has proved that in the case when the bigsize of a squarefree monomial ideal is 2, Stanley's conjecture holds. One of the key reasons for this is that in this particular case the depth of the squarefree monomial ideal does not depend on the characteristic of the base field. The next natural step we would like to make is when the bigsize of the squarefree monomial ideal is 3. We have already some partial results in this direction, but the main difficulty we have to deal with is that the depth may depend on the characteristic.
A secondary goal of the project is related to a better understanding of the algebraic invariants of the binomial edge ideals in terms of the combinatorial properties of the underlying graph and, more generally, of the algebraic invariants of determinantal ideals associated with simplicial complexes in terms of the combinatorial properties of the complex. This is a very hard problem in general, but in some particular cases one can give nice descriptions. For example, recently it was shown by Herzog&al. that the depth of a binomial edge ideal corresponding to a forest can be computed only from the graph's data, and consequently a description of the Cohen-Macaulay property can be given in terms of graph theory. Furthermore, they show that for closed graphs the Cohen-Macaulay property of the binomial edge ideal J is equivalent to the Cohen-Macaulay property of the initial ideal in(J) and in this case algebraic invariants like Hilbert function and multiplicity can be easily computed. Of an utmost importance in stating and proving the conjectures regarding binomial edge ideals is the extensive use of the computer algebra packages CoCoA and Singular. Consequently, one of the problems we plan to attack, at least for particular classes of graphs, was conjectured after extensive computations and states that for all graphs G the extremal Betti numbers of J_G and in(J_G) coincide.
UpProject director
C.S.I. Dr. Dorin-Mihail POPESCU, "Simion Stoilow" Institute of Mathematics of the Romanian Academy, Bucharest (IMAR) .The research team of the project
Objectives and expected results: [txt]
Scientific results obtained in the project
Activities