ALGEBRAIC MODELLING OF SOME COMBINATORIAL OBJECTS

AND COMPUTATIONAL APPLICATIONS

PROGRAM PN II-HUMAN RESOURCES, CODE TE_46, contract nr. 83/02.08.2010

Brief description of the research project

A topic of large interest in the current mathematical research is formed by two conjectures formulated by Richard Stanley. One of them refers to the so called Stanley decompositions of finitely generated multigraded modules over a polynomial ring in n variables (with the standard grading). The second conjecture refers to partitionable simplicial complexes.
The former was proposed by Stanley in 1982 in his famous paper Linear Diophantine equations and local cohomology published by Inventiones Mathematicae. For 23 years, the statement was validated in only few isolated cases. This conjecture states that any finitely generated multigraded module over the standard graded polynomial ring in several variables admits a Stanley decomposition whose Stanley depth (sdepth) is bounded from below by the depth of the module. It has been recently shown that the first conjecture implies the second conjecture that was mentioned above.

In the following, we shall refer only to the first conjecture of Stanley. Currently, it is widely open. It is known to hold for polynomial rings in at most 5 variables and in several other cases. One of the reasons is that it is somehow unclear what is the connection between depth (a homological invariant) and sdepth (a combinatorial invariant). Lacking a general strategy, the methods that have been employed so far relied very much on the particularities of the respective situations. In this context, the paper How to compute the Stanley depth of a monomial ideal , Journal of Algebra (Volume 322 (9), 2009, pp. 3151-3169) of Herzog, Vladoiu and Zheng brings a general strategy for tackling the problem. Furthermore, it is given an effective algorithm for computing sdepth of modules of the form J/I, where J and I are arbitrary monomial ideals of a polynomial ring. This new approach allowed for the verification of the conjecture in several new cases, besides the class of pretty clean modules. It is worth mentioning that for pretty clean modules of the form J/I with J and I monomial ideals in a polynomial ring, the equality depth J/I = sdepth J/I holds. The new strategy allowed the study of the general case when depth and sdepth are different. Moreover, the previous known cases could be given a more elegant and simpler proof and they became easier accessible to the interested audience. The above paper of Herzog et al had a large impact and rather quickly. It has already been cited in 35 scientific papers in the field of commutative algebra, combinatorics and computational algebra.

As a natural consequence of the insterest that was raised by the paper in the scientific community, one of the main objectives of this project is the theoretical refinement of the algorithm for computing sdepth of a monomial ideal. We also aim at extending it for finitely generated multigraded modules over a polynomial ring with the standard grading. We want to produce an efficient implementation of the algorithm in one of the specialised computer algebra softwares like Singular, CoCoA or Macaulay2. That would help us verify Stanley's conjecture in polynomial rings with a larger number of variables. We hope to obtain new classes of ideals for which that conjecture holds.

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The research team of the grant

Lect. Dr. Vladoiu Grigore-Marius, FMI-UB: CV [pdf] , Citation. [pdf]

C.P. III Dr. Ichim Bogdan, IMAR: CV [pdf]

Asist. Dr. Stamate Dumitru, FMI-UB: CV [pdf]

Asist. cerc. Drd. Epure Mihai, IMAR, FMI-UB: CV [pdf]

Reports (in Romanian)

2010 Term: 10.12.2010

• Raport anual de activitate
• Sinteza lucrarii
• Deviz postcalcul

2011 Term: 10.12.2011

• Raport anual de activitate
• Sinteza lucrarii
• Deviz postcalcul

2012 Term: 5.12.2012

• Raport anual de activitate
• Sinteza lucrarii
• Deviz postcalcul

2013 Term: 25.09.2013

• Raport anual de activitate
• Sinteza lucrarii
• Deviz postcalcul

Final audit report of the budget of the grant. [pdf]

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Results obtained

1. 2010:
   • Juergen Herzog, Dorin Popescu, Marius Vladoiu - Stanley depth and size of a monomial ideal, Proceedings of the American Mathematical Society, vol. 140 (2012), 493-504. DOI 10.1090/S0002-9939-2012-11371-1. Preprint version arXiv:1011.6462 [math.AC].
   • Mircea Becheanu, Marius Vladoiu - Irreducibility criteria for polynomials via Jordan matrices. Submitted.
   • W. Bruns, B. Ichim, C. Soeger - Normaliz 2.5. Available at http://www.math.uos.de/normaliz.
   • V. Almendra, B. Ichim - jNormaliz. A graphical interface for Normaliz. Available at http://www.math.uos.de/normaliz.
2. 2011:
   • Juergen Herzog, Asia Rauf, Marius Vladoiu - The stable set of associated prime ideals of a polymatroidal ideal, J. Alg. Combinatorics, Vol. 37(2), 2013, 289-312. DOI:10.1007/s10801-012-0367-z. Preprint version arXiv:1109.5834 [math.AC].
   • Zaheer Ahmad, Tiberiu Dumitrescu, Mihai Epure - A Schreier Domain Type Condition, Bull. Math. Soc. Sci. Math. Roumanie, vol. 55(3), 2012, 241-247. Preprint version arXiv:1112.0132 [math.AC].
   • Zaheer Ahmad, Tiberiu Dumitrescu, Mihai Epure - A Schreier Domain Type Condition II, Algebra Colloq. 22 (2015), 923-924. DOI: 10.1142/S1005386715000772 . Preprint version arXiv:1112.1236 [math.AC].
   • W. Bruns, B. Ichim, C. Soeger - Normaliz 2.7. Disponibil la http://www.math.uos.de/normaliz.
3. 2012:
   • Juergen Herzog, Marius Vladoiu - Squarefree monomial ideals with constant depth function, J. Pure Applied Algebra, Vol 217(9), 2013, 1764-1772. DOI:10.1016/j.jpaa.2012.12.009. Preprint version arXiv:1209.5890 [math.AC].
   • Bogdan Ichim, Julio Moyano - How to compute the multigraded Hilbert depth of a module, Mathematische Nachrichten, Volume 287, Issue 11-12, 2014, Pages 1274-1287. DOI: 10.1002/mana.201300120. Preprint version arXiv:1209.0084 [math.AC].
   • W. Bruns, B. Ichim, C. Soeger - Normaliz 2.8. Disponibil la http://www.math.uos.de/normaliz.
   • V. Almendra, B. Ichim - jNormaliz 1.2. A graphical interface for Normaliz. Available at http://www.math.uos.de/normaliz.
   • Mihai Epure - A Schreier domain type condition in the ideal systems setting. Submitted.
4. 2013:
   • H. Charalambous, A. Thoma, M. Vladoiu - Markov Bases of Lattice Ideals, Preprint 2013 (22pp). arXiv:1303.2303 [math.AC].
   • H. Charalambous, A. Thoma, M. Vladoiu - Markov bases and generalized Lawrence liftings, Annals of Combinatorics 19 (2015), 661-669. DOI: 10.1007/s00026-015-0287-4. Preprint version arXiv:1304.4257 [math.AC].
   • D. I. Stamate - Asymptotic properties in the shifted family of a numerical semigroup with few generators, Semigroup Forum 93 (2016), 225-246. DOI 10.1007/s00233-015-9724-2.
   • M. Rosca, M. Vladoiu -Weighted vertex cover algebras, in preparation.

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Conference talks/Dissemination of results

Date	Title of the talk - author
14.09.2010	Normaliz 2.5. - Bogdan Ichim, ICMS 2010 International Congress on Mathematical Software, Kobe, Japonia
20.09.2010	Polyhedra and their Faces - Marius Vladoiu, SNA- Combinatorics in Commutative Algebra, IMAR
21.09.2010	Finite Generation of Cones - Marius Vladoiu, SNA- Combinatorics in Commutative Algebra, IMAR
22.09.2010	Affine Monoids and their Hilbert Bases- Marius Vladoiu, SNA- Combinatorics in Commutative Algebra, IMAR
23.09.2010	Affine Monoid Rings - Bogdan Ichim, SNA- Combinatorics in Commutative Algebra, IMAR
24.09.2010	Normal Affine Monoid Rings - Bogdan Ichim, SNA- Combinatorics in Commutative Algebra, IMAR
24.09.2010	Introduction to Normaliz - Bogdan Ichim, SNA- Combinatorics in Commutative Algebra, IMAR
13.02.2011	Stanley depth and size of a monomial ideal - Marius Vladoiu, 5thWorld Conference on 21st Century Mathematics 2011, Lahore, Pakistan
13.05.2011	Koszul numerical semigroups - Dumitru Stamate, WYRM, Constanta
22.06.2011	Introduction to Normaliz 2.7 - Bogdan Ichim, MMMA 2011(Matrix Methods in Mathematics and Applications), Moscow, Russia
29.06.2011	Introduction to Normaliz 2.7 - Bogdan Ichim, 7th Congress of Romanian Mathematicians, Brasov
11.07.2011	Stanley depth and size of a monomial ideal - Marius Vladoiu, MONomial Ideals, Computations and Applications,CIEM Castro Urdiales (Cantabria, Spania)
19.09.2011	Affine monoids and Hilbert bases I - Marius Vladoiu, SNA- Computer Algebra and Combinatorics, IMAR
20.09.2011	Affine monoids and Hilbert bases II - Marius Vladoiu, SNA- Computer Algebra and Combinatorics, IMAR
20.09.2011	Hilbert functions and Ehrhart functions I - Bogdan Ichim, SNA- Computer Algebra and Combinatorics, IMAR
21.09.2011	Hilbert functions and Ehrhart functions II - Bogdan Ichim, SNA- Computer Algebra and Combinatorics, IMAR
22.09.2011	Computing convex hulls and triangulations - Marius Vladoiu SNA- Computer Algebra and Combinatorics, IMAR
20.10.2011	The Koszul property for affine semigroups - Dumitru Stamate, International School ISCCAAG, Messina, Italia
9.05.2012	Introduction to Normaliz - Bogdan Ichim, Universitatea Rostock, Germania
11.05.2012	The stable set of associated prime ideals of a polymatroidal ideal - Marius Vladoiu, WYRM 2012, Universitatea Ovidius, Constanta
11.05.2012	Semigroups with few generators and shellings - Dumitru Stamate, WYRM 2012, Universitatea Ovidius, Constanta
11.05.2012	A Schreier Domain Type Condition - Mihai Epure, WYRM 2012, Universitatea Ovidius, Constanta
4.09.2012	Shellings for semigroups - Dumitru Stamate, SNA-Discrete Invariants in Commutative Algebra, Mangalia
4.09.2012	A new class of pseudo-Dedekind domains and its star operations extensions - Mihai Epure, SNA-Discrete Invariants in Commutative Algebra, Mangalia
4.09.2012	Resolutions and Singular-Tutorial - Dumitru Stamate, SNA-Discrete Invariants in Commutative Algebra, Mangalia
7.09.2012	Rational polytopes in combinatorial voting theory - Bogdan Ichim, SNA-Discrete Invariants in Commutative Algebra, Mangalia
20.11.2012	How to compute the multigraded Hilbert depth of a module - Bogdan Ichim, Universitatea Osnabrueck, Germania
21.12.2012	A Schreier domain type condition and its star operation extensions - Mihai Epure, Commutative rings, integer-valued polynomials and polynomial functions, Graz, Austria
8.1.2013	A Schreier-type condition domain in the star operation setting (II) - Mihai Epure, Comm. Algebra Seminar, IMAR & FMI, Bucharest
15.1.2013	A Schreier-type condition domain in the star operation setting (III) - Mihai Epure, Comm. Algebra Seminar, IMAR & FMI, Bucharest
26.02.2013	The Nagata ring of a t-sharp domain - Mihai Epure, Comm. Algebra Seminar, IMAR & FMI, Bucharest
12.03.2013	The Complete Intersection property for shifted semigroups - Dumitru Stamate, Comm. Algebra Seminar, IMAR & FMI, Bucharest
02.04.2013	The Complete Intersection property for shifted semigroups (II)- Dumitru Stamate, Comm. Algebra Seminar, IMAR & FMI, Bucharest
28.05.2013	A Schreier domain type condition in the systems of ideals context - Mihai Epure, Comm. Algebra Seminar, IMAR & FMI, Bucharest
27.06.2013	Behaviour of depth function for monomial ideals - Marius Vladoiu, EACA's Second International School On Computer Algebra and Applications, Valladolid, Spain
05.09.2013	A Schreier domain type condition, its star operation extensions and a more general setting (ideal systems for monoids) - Mihai Epure, SNA-Algebraic Methods in Combinatorics, IMAR, Bucharest