Grant PN-II-ID-PCE-2011-3-1023 Algorithmic and theoretical methods for studying monomial and binomial ideals with applications in combinatorics, commutative algebra and graph theory

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.

Project 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

C.S.I Dr. Dorin-Mihai Popescu, IMAR: CV [pdf] , List of publications [pdf]
Prof. Dr. Viviana Ene , Univ Ovidius-Constanta, IMAR : CV [pdf]
C.S. III Dr. Bogdan Ichim, IMAR: CV [pdf]
Lect. Dr. Marius Vladoiu, University of Bucharest (FMI-UB), IMAR: CV [pdf] ,
C.S. Dr. Mircea Cimpoeas, IMAR: CV [pdf]
Asist. Dr. Dumitru Stamate, FMI-UB, IMAR: CV [pdf]
Dr. Andrei Zarojanu, FMI-UB, IMAR: CV [pdf]
Miruna Rosca, FMI-UB, IMAR: CV [pdf]

Budget

Duration of the grant: 01.01.2012-31.12.2016
Financing institution: Unitatea Executiva pentru Finantarea Invatamantului Superior, a Cercetarii, Dezvoltarii si Inovarii, Romania (UEFISCDI)
Host institution: Simion Stoilow Institute of Mathematics of the Romanian Academy, Bucharest, Romania (IMAR)

Objectives and expected results: [txt]

Reports

2012

2013

2014

2015

2016

Scientific results obtained in the project

A. Aslam, V. Ene, Simplicial complexes with rigid depth, Arch. Math. 99 (4) (2012), 315-325. Preprint version available at arXiv:1201.3325 [math.AC]
M. Cimpoeas, The Stanley conjecture on monomial almost complete intersection ideals, Bull. Soc. Math. Roumanie, vol 55 (103), no. 1 (2012), 35-39.
M. Cimpoeas, Several inequalities regarding Stanley depth, Romanian Journal of Mathematics and Computer Science , vol 2, no. 1 (2012), 28-40.
M. Cimpoeas, Stanley depth of quotient of monomial complete intersection ideals, Communications in Algebra,Volume 42, Issue 10 (2014) 4274-4280. Preprint version available at arXiv:1210.2214 [math.AC].
V. Ene, J. Herzog, T. Hibi, F. Mohammadi, Determinantal facet ideals, Michigan Math. J., Volume 62, Issue 1 (2013), 39-57.
V. Ene, A. Qureshi, Ideals generated by diagonal 2-minors, Communications in Algebra, Volume 41, Issue 8 (2013), 3058-3066.
V. Ene, R. Okazaki, On the radical of multigraded modules, Journal of Algebra, Volume 388, 10-21. Preprint version available at arXiv:1210.2026[math.AC].
D. Popescu, Depth of factors of square free monomial ideals, Proceedings of the American Mathematical Society, Volume 142 (2014), 1965-1972.
D. Popescu, Upper bounds of depth of monomial ideals, J. Commutative Alg., vol 5, no 2 (2013), 323-327. Preprint version available at arXiv:1206.3977 [math.AC].
D. Popescu, A.Zarojanu, Depth of some square free monomial ideals, Bull. Math. Soc. Sci. Math. Roumanie, vol 56 (104), no. 1 (2013), 117-124. Preprint version available at arXiv:1211.0842 [math.AC].
D. Popescu, A.Zarojanu, Depth of some special monomial ideals, Bull. Math. Soc. Sci. Math. Roumanie, vol 56 (104), 2013, 365-368. Preprint version available at arXiv:1301.5171 [math.AC].
M. Cimpoeas, Vertex cover algebras of simplicial multicomplexes, Romanian Journal of Mathematics and Computer Science , vol 3, no. 1 (2013), 1-4 .
B. Ichim, A. Zarojanu, An algorithm for computing the multigraded Hilbert depth of a module, Experimental Mathematics 23 (3), 322-331, 2014. Preprint version available at arXiv:1304.7215 [math.AC] .
V. Ene, A.Zarojanu, On the regularity of binomial edge ideals, Mathematische Nachrichten, Volume 288, Issue 1 (2015), 19-24. Preprint version available at arXiv:1307.2141 [math.AC] .
V. Ene, A. A. Qureshi, A. Rauf, Regularity of join-meet ideals of distributive lattices, Electron J. Combin. vol 20 (3) (2013) #P20. Preprint version available at arXiv:1307.7557 [math.AC] .
D. Popescu, A. Zarojanu, Three generated, squarefree, monomial ideals, Bull. Math. Soc. Sci. Math. Roumanie, vol 58 (106), no. 3 (2015), 359-368. Preprint version available at arXiv:1307.8292 [math.AC] .
H. Charalambous, A. Thoma, M. Vladoiu, Markov complexity of monomial curves. J. Algebra, vol 417 (2014), 391-411. Preprint version available at arXiv:1311.4707 [math.AC].
J. Herzog, M. Vladoiu, Monomial ideals with primary components given by powers of monomial prime ideals, The Electronic J. Combinatorics vol 21 (1) (2014), #P1.69. Preprint version available at arXiv:1310.3409 [math.AC].
D. Popescu, Stanley depth on five generated, squarefree, monomial ideals, Bull. Math. Soc. Sci. Math. Roumanie, vol 59 (107), no. 1 (2016), 75-99. Preprint version available at arXiv:1312.0923 [math.AC].
V. Ene, J. Herzog, T. Hibi, Koszul binomial edge ideals, Bridging Algebra, Geometry, and Topology, Springer Proceedings in Mathematics & Statistics, 96, D. Ibadula, W. Veys (Eds.) Springer, 2014, 127-138. Preprint version available at arXiv:1310.6426 [math.AC].
V. Ene, J. Herzog, T. Hibi, Linear flags and Koszul filtrations, Kyoto J. Math., vol. 55, no.3 (2015), 517-530. Preprint version available at arXiv:1312.2190 [math.AC].
V. Ene, J. Herzog, T. Hibi, Linearly related polyominoes, J. Algebraic Combin., Volume 41 (2015), Issue 4, 949-968. Preprint version available at arXiv:1403.4349 [math.AC].
A. Dimca, D. Popescu, Hilbert series and Lefschetz properties of dimension one almost complete intersections, Communications in Algebra 44 (2016), 4467-4482. Preprint version available at arXiv:1403.5921 [math.AG].
V. Ene, J. Herzog, S. Saeedi Madani, A note on the regularity of the Hibi rings, Manuscripta Mathematica, 148(3) (2015), 501-506. Preprint version available at arXiv:1404.2554 [math.AC].
F. Chaudhry, A. Dokuyucu, V. Ene, Binomial edge ideals and rational normal scrolls, Bull. Iranian Math. Soc., Vol. 41 (2015), No. 4, 971-979. Preprint version available at arXiv:1404.7602 [math.AC].
V. Ene, J. Herzog, T. Hibi, S. Saeedi Madani, Pseudo-Gorenstein and level Hibi rings, Journal of Algebra, vol 431 (2015), 138-161. Preprint version available at arXiv:1405.6963 [math.AC].
D. Popescu, Depth in a pathological case, Bull. Math. Soc. Sci. Math. Roumanie, vol 59 (107), no. 2 (2016), 187-195. Preprint version available at arXiv:1406.1398 [math.AC].
V. Ene, Syzygies of Hibi rings, Acta Mathematica Vietnamica, Special Issue on: Commutative Algebra and its Interaction with Algebraic Geometry and Combinatorics II 40 (2015) no. 3, 403-446. Special volume dedicated to the 60th birthday of Professor NV Trung. Preprint version available at arXiv:1409.2445 [math.AC].
Mircea Cimpoeas, Dumitru I. Stamate, On intersections of complete intersection ideals, Journal of Pure and Applied Algebra, vol 220, no. 11 (2016), 3702-3712. Preprint 12 pp.
D. Popescu, Around General Neron Desingularization, to appear in Journal of Algebra and Its Applications, 16, No. 2 (2017). Preprint version available at arXiv:1504.06938 [math.AC].
A. Popescu, D. Popescu, A method to compute the General Neron Desingularization in the frame of one dimensional local domains, to appear in Singularities and Computer Algebra- Festschrift for Gert-Martin Greuel on the Occasion of his 70th Birthday, Editors Wolfram Decker, Gerhard Pfister, Mathias Schulze, Springer Monograph. Preprint version available at arXiv:1508.05511 [math.AC].
D. Popescu, Artin approximation property and the General Neron Desingularization, to appear in Revue Roum. Math. Pures et Appl., 2017. Preprint version available at arXiv:1511.06967 [math.AC].
M. Cimpoeas, Stanley depth of the path ideal associated to a line graph, to appear in Math. Reports vol. 19, no. 2 (2017). Preprint version available at arXiv:1508.07540 [math.AC].
M. Cimpoeas, On the quasi-depth of squarefree monomial ideals and the sdepth of the monomial ideal of independent sets of a graph. Preprint version available at arXiv:1511.06974 [math.AC].
G. Pfister, D. Popescu, Constructive General Neron Desingularization for one dimensional local rings, J. of Symbolic Computation vol. 80 (2017), 570-580. Preprint version available at arXiv:1512.08435 [math.AC].
M. Cimpoeas, On the Stanley depth of powers of some classes of monomial ideals, 8 pp, to appear in Bulletin of the Iranian Mathematical Society. Preprint version available at arXiv:1512.08195 [math.AC].
M. Cimpoeas, On the Stanley depth of the path ideal of a cycle graph, Romanian Journal of Mathematics and Computer Sciences, vol 6, no. 2 (2016), 116-120. Preprint version available at arXiv:1601.00261 [math.AC].
D. Popescu, Nested Artin approximation. Preprint version available at arXiv:1601.06654 [math.AC].
F. J. Castro-Jimenez, D. Popescu, G. Rond, Linear nested Artin approximation for algebraic power series. Preprint version available at arXiv:1511.09275 [math.AC].
D. Popescu, Size and Stanley depth of monomial ideals. Preprint version available at arXiv:1602.06760 [math.AC].
M. Cimpoeas, On the Stanley depth of a special class of Borel type ideals, 7 pp, submitted. Preprint version available at arXiv:1603.03939 [math.AC].
M. Cimpoeas, A class of square-free monomial ideals associated to two integer sequences, 15 pp, submitted. Preprint version available at arXiv:1604.02933 [math.AC].
M. Cimpoeas, F. Nicolae, On the restricted partition function, 21 pp, submitted. Preprint version available at arXiv:1609.06090 [math.CO].
M. Cimpoeas, F. Nicolae, On the restricted partition function, II, 14 pp. Preprint version available at arXiv:1611.00256 [math.CO].
M. Cimpoeas, D. Stamate, Groebner-nice pairs of ideals, in preparation.

Activities

the members of the team are regular speakears of the weekly Commutative Algebra seminar run jointly by IMAR and FMI-UNIBUC, Tuesdays, 12-14, room 120 at FMI-UNIBUC.
D. Popescu, V. Ene, B. Ichim, M. Cimpoeas and D. Stamate were among the organizers of the 20th National School on Algebra, 2-8 Sept 2012, Mangalia.
V. Ene, Groebner Basics, tutorial at the 20th National School on Algebra, 3 Sept 2012, Mangalia.
D. Stamate, Shellings for semigroups, talk at the 20th National School on Algebra, 4 Sept 2012, Mangalia.
B. Ichim, Rational polytopes in combinatorial voting theory, talk at the 20th National School on Algebra, 7 Sept 2012, Mangalia.
B. Ichim, Introduction to Normaliz, talk at Rostock University, May 9, 2012.
B. Ichim, How to compute the multigraded Hilbert depth of a module, talk at Osnabrueck University, Germany, 20 Nov 2012.
D. Popescu, New results and contributions towards a proof of Stanley's Conjecture, talk at University of Kaiserslautern, Germany, July 2012.
Viviana Ene- research visit at the University of Essen, collaboration with prof. Juergen Herzog, Oct 2012.
D. Popescu, Recent results on Stanley's Conjecture, talk at the Workshop on Rings, Categories and Hopf Algebra, Univ of Bucharest, 18-19 May 2013.
D. Popescu, A hope for the Stanley Conjecture on monomial ideals, talk at the international conference Experimental and Theoretical Methods in Algebra, Geometry and Topology, Eforie Nord, 21-24 June 2013.
V. Ene, Binomial ideals and graphs, talk at the Joint International Meeting of the American Mathematical Society and the Romanian Mathematical Society, Alba Iulia, 27-30 June 2013.
D. Popescu, Around Stanley's conjecture on monomial ideals, talk at the Anniversary Conference Faculty of Sciences - 150 years, University of Bucharest, 28 Aug-1 Sep 2013.
V. Ene, Binomial edge ideals, talk at the Anniversary Conference Faculty of Sciences - 150 years, University of Bucharest, 28 Aug-1 Sep 2013.
V. Ene, M. Rosca, D. Stamate, M. Vladoiu and A. Zarojanu, together with Mihai Epure organized the 21st Edition of the National Algebra School- Algebraic Methods in Combinatorics, Institute of Mathematics, Bucharest, 2-6 Sept 2013.
D. Popescu and M. Vladoiu were part of the Scientific committee of the school, together with T. Roemer, Univ. of Osnabrueck, Germany.
V. Ene, Algebraic and homological properties of binomial edge ideals, 2 talks and 1 tutorial at SNA-2013, Bucharest, 2-6 Sept 2013.
D. Stamate, Tools of Combinatorial Commutative Algebra, talk at SNA-2013, Bucharest, 2-6 Sept 2013.
D. Stamate, Matroids and realisability, talk at SNA-2013, Bucharest, 2-6 Sept 2013.
M. Vladoiu, Polymatroidal ideals, 2 talks and a tutorial at SNA-2013, Bucharest, 2-6 Sept 2013.
M. Cimpoeas, Stanley depth of quotient of monomial complete intersection ideals, talk at SNA-2013, Bucharest, 2-6 Sept 2013.
A. Zarojanu, Depth of some special monomial ideals, talk at SNA-2013, Bucharest, 2-6 Sept 2013.
V. Ene, M. Rosca, D. Stamate, M. Vladoiu and A. Zarojanu, together with Florin Ambro and Mihai Epure organized the 22nd Edition of the National Algebra School - Algebraic and Combinatorial Applications of Toric Ideals, Institute of Mathematics, Bucharest, 1-5 Sept 2014.
Viviana Ene, Hibi rings and their Groebner bases, talk at SNA-2014, Bucharest, 1-5 Sept 2014.
Miruna Rosca, Vertex cover algebras of weighted graphs, talk at SNA-2014, Bucharest, 1-5 Sept 2014.
Viviana Ene, Level and pseudo - Gorenstein Hibi rings, talk at SNA-2014, Bucharest, 1-5 Sept 2014.
Andrei Zarojanu, An algorithm for computing the multigraded Hilbert depth of a module, talk at SNA-2014, Bucharest, 1-5 Sept 2014.
Viviana Ene, The regularity of Hibi rings, talk at SNA-2014, Bucharest, 1-5 Sept 2014.
Marius Vladoiu, Bouquet algebra of toric ideals, a series of 4 talks at the Commutative Algebra seminar IMAR and Univ. Bucuresti, 6 January, 10 March, 7 April, 21 April 2015.
Mircea Cimpoeas, On intersections of complete intersection ideals, talk at the Commutative Algebra seminar IMAR and Univ. Bucuresti, 24 February 2015.
Dorin Popescu, Smooth morphisms, talk at the Thematic Semester on Artin Approximation and Singularity Theory 2015, Luminy, Spring 2015.
Dorin Popescu, The smooth locus of a morphism, talk at the Thematic Semester on Artin Approximation and Singularity Theory 2015, Luminy, Spring 2015.
Dorin Popescu, Elkik's theorem, talk at the Thematic Semester on Artin Approximation and Singularity Theory 2015, Luminy, Spring 2015.
Dorin Popescu, General Neron Desingularization in dimension 1, talk at the Thematic Semester on Artin Approximation and Singularity Theory 2015, Luminy, Spring 2015.
Dorin Popescu, Artin approximation, versal deformations, and maximal Cohen-Macaulay modules , talk at the Thematic Semester on Artin Approximation and Singularity Theory 2015, Luminy, Spring 2015.
Dorin Popescu, On regular local rings, talk at the Thematic Semester on Artin Approximation and Singularity Theory 2015, Luminy, Spring 2015.
Dorin Popescu, Around General Neron Desingularization, talk at the Commutative Algebra seminar IMAR and Univ. Bucuresti, 31 March 2015.
Dorin Popescu, A theorem of Ploski's type, talk at the Special Session on Algebra in the memory of Prof. Serban Basarab, Univ. Ovidius, Constanta, 22 May 2015.
Dorin Popescu, together with S. Dascalescu, A. Marcus, D. Stefan and A. Zaharescu formed the scientific committee of the general Section on Algebra and Number Theory, at the 8th Congress of Romanian Mathematicians, Iasi, Romania, June 26- July 2, 2015.
Dorin Popescu and Dumitru Stamate organized the Special session: Local rings and homological algebra, dedicated to Prof. Nicolae Radu, at the 8th Congress of Romanian Mathematicians, Iasi, Romania, 27 June 2015.
Viviana Ene, Ideals of 2-minors, talk at The Eighth Congress of Romanian Mathematicians, Iasi, June 26- July 2, 2015.
Dorin Popescu, A theorem of Ploski's type, talk at The Eighth Congress of Romanian Mathematicians, Iasi, June 26- July 2, 2015.
Marius Vladoiu, Bouquet Algebra of Toric Ideals, talk at The Eighth Congress of Romanian Mathematicians, Iasi, June 26- July 2, 2015.
Andrei Zarojanu, On the Stanley Depth, talk at The Eighth Congress of Romanian Mathematicians, Iasi, June 26- July 2, 2015.
V. Ene, D. Stamate, M. Vladoiu, together with Mihai Epure organized the 23rd Edition of the National Algebra School - Interactions of Computer Algebra with Commutative Algebra, Combinatorics and Algebraic Statistics , Institute of Mathematics, Bucharest, 31 Aug - 4 Sept 2015.
Scientific committee: Gerhard Pfister (U. Kaiserslautern) and Dorin Popescu.
Mircea Cimpoeas, Periodicity results for intersections of toric ideals, talk at SNA-2015, Bucharest, 31 Aug - 4 Sept 2015.
Viviana Ene, Quadratic Groebner bases and Koszul filtrations, 2 lectures at SNA-2015, Bucharest, 31 Aug - 4 Sept 2015.
Marius Vladoiu, Geometric Theorem Proving, talk at SNA-2015, Bucharest, 31 Aug - 4 Sept 2015.
Viviana Ene, Koszul binomial edge ideals of pairs of graphs, talk at the Commutative Algebra seminar IMAR and Univ. Bucuresti, 27 October 2015.
Dorin Popescu, Around General Neron Desingularization , talk at Leuven, 4 November 2015.
Dumitru Stamate, An introduction to the Eisenbud-Green-Harris Conjecture(s), talk at the Commutative Algebra seminar IMAR and Univ. Bucuresti, 17 November 2015.
Dumitru Stamate, Quadratic numerical semigroups and the Koszul property, talk at the Welcome Workshop University of Genova, Italy, 2 February 2016.
Mircea Cimpoeas, Stanley depth of some classes of monomial ideals, talk at the Commutative Algebra seminar IMAR and Univ. Bucuresti, 15 and 22 March 2016.
Mircea Cimpoeas, Stanley depth of a class of square-free monomial ideals, talk at the Workshop for Young Researchers in Mathematics Constanta, Romania, 19-22 mai 2016.
Dumitru Stamate, Periodic properties in the shifted family of a numerical semigroup, talk at the Workshop for Young Researchers in Mathematics Constanta, Romania, 19-22 mai 2016.
Mircea Cimpoeas, On intersection of complete intersection ideals, talk at the International meeting on numerical semigroups with applications , Levico Terme, Italy, 4-8 July 2016.
Dumitru Stamate, Quadratic numerical semigroups and the Koszul property, talk at the International meeting on numerical semigroups with applications , Levico Terme, Italy, 4-8 July 2016.
V. Ene, D. Stamate, M. Vladoiu organized the 24th Edition of the National Algebra School - EMS Summer School on Multigraded Algebra and Applications with support from Foundation Compositio Mathematica , Moieciu de Sus, 17 Aug - 24 Aug 2016.
Scientific committee: Viviana Ene (Ovidius University, Romania), Juergen Herzog (Duisburg-Essen University, Germany), Thomas Kahle (Magdeburg University, Germany), Ezra Miller (Duke University, USA), Apostolos Thoma (Ioannina University, Greece).
Dumitru Stamate, Quadratic numerical semigroups and the Koszul property, talk at the Algebra Seminar, University Osnabrueck, Germany, 27 September 2016.
Mircea Cimpoeas, On the restricted partition function, talk at the Commutative Algebra seminar IMAR and Univ. Bucuresti, 18 October 2016.
Dumitru Stamate, Betti numbers for numerical semigroup rings, talk at the Workshop and Conference on Commutative Algebra , Tribhuvan University, Kirtipur, Kathmandu, Nepal, 20 Oct 2016.
Dumitru Stamate, On the Koszul property for numerical semigroup rings, talk at the Workshop and Conference on Commutative Algebra , Tribhuvan University, Kirtipur, Kathmandu, Nepal, 24 Oct 2016.