MODELAREA ALGEBRICA A UNOR OBIECTE COMBINATORIALE

SI APLICATII COMPUTATIONALE

PROGRAM PN II-RESURSE UMANE, COD TE_46, contract nr. 83/02.08.2010

Descriere succinta a proiectului de cercetare

Un subiect de mare interes in cercetarea matematica actuala il constituie doua conjecturi formulate de Richard Stanley. Prima din ele se refera la asa-numitele descompuneri Stanley ale modulelor multigraduate finit generate peste inele de polinoame in n variabile (cu graduarea standard). A doua conjectura se refera la complexele simpliciale partitionabile.
Prima conjectura a fost lansata de Stanley in 1982 intr-un articol faimos Linear Diophantine equations and local cohomology aparut in Inventiones Mathematicae. Timp de 23 de ani ea a fost validata doar in cateva cazuri izolate. Aceasta conjectura afirma ca orice modul multigraduat finit generat peste inelul de polinoame in mai multe variabile standard graduat admite o descompunere Stanley al carei Stanley depth (sdepth) este marginit inferior de depth-ul modulului. Recent a fost demonstrat ca prima conjectura mentionata implica de fapt a doua conjectura.

In ceea ce urmeaza ne vom referi doar la prima conjectura formulata de Stanley. Conjectura este larg deschisa. Ea a fost verificata pentru inele de polinoame cu cel mult 5 variabile si in alte cateva cazuri. Legatura dintre depth (un invariant omologic) si sdepth (un invariant combinatorial) este oarecum neclara. In lipsa unei strategii globale, metodele folosite au depins foarte mult de particularitatile cazurilor tratate. In acest context, lucrarea How to compute the Stanley depth of a monomial ideal , Journal of Algebra (Volume 322 (9), 2009, pp. 3151-3169) scrisa de Herzog, Vladoiu si Zheng propune o strategie generala pentru abordarea acestei problemei. Mai mult, este prezentat si un algoritm de calcul efectiv pentru sdepth in cazul unui modul de forma J/I, unde J, I sunt ideale monomiale arbitrare intr-un inel de polinoame. Aceasta noua abordare a permis verificarea conjecturii pentru noi cazuri, in afara clasei modulelor pretty clean. Merita mentionat ca in clasa modulelor pretty clean de forma J/I cu J si I ideale monomiale intr-un inel polinomial are loc egalitatea depth J/I = sdepth J/I. Noua strategie a permis si studierea cazului general, in care depth si sdepth sunt diferite. De asemenea, cazurile anterior cunoscute au putut fi redemonstrate intr-o maniera mai simpla si mai eleganta devenind astfel mai usor accesibile celor interesati. Impactul articolului in comunitatea matematica a fost rapid si destul de mare. El a fost deja citat in 35 de lucrari stiintifice din domeniul algebrei comutative, combinatoricii si al algebrei computationale.

Ca o consecinta naturala a interesului starnit in comunitatea stiintifica de lucrarea mai sus mentionata, unul din obiectivele principale ale acestui proiect este rafinarea teoretica a algoritmului de calcul al sdepth-ului unui ideal monomial. Totodata vizam si generalizarea lui avand drept tinta cazul general al modulele multigraduate finit generate peste un inel de polinoame in n variabile standard graduat. Dorim de asemenea sa furnizam o implementare eficienta a algoritmului intr-un sistem specializat de algebra computationala cum ar fi Singular, CoCoA ori Macaulay2 si sa folosim noul software pentru a verifica conjectura Stanley pentru un numar mai mare de variabile. Nu in ultimul rand, ne dorim sa obtinem noi cazuri favorabile in care aceasta conjectura este adevarata.

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Echipa de cercetare a grantului

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

Rapoarte

2010 Etapa unica: 10.12.2010

• Raport anual de activitate
• Sinteza lucrarii
• Deviz postcalcul

2011 Etapa unica: 10.12.2011

• Raport anual de activitate
• Sinteza lucrarii
• Deviz postcalcul

2012 Etapa unica: 5.12.2012

• Raport anual de activitate
• Sinteza lucrarii
• Deviz postcalcul

2013 Etapa unica: 25.09.2013

• Raport anual de activitate
• Sinteza lucrarii
• Deviz postcalcul

Raport de audit privind bugetul grantului. [pdf]

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Rezultate obtinute

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. Trimis spre publicare.
   • W. Bruns, B. Ichim, C. Soeger - Normaliz 2.5. Disponibil la http://www.math.uos.de/normaliz.
   • V. Almendra, B. Ichim - jNormaliz. A graphical interface for Normaliz. Disponibil la 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. Disponibil la http://www.math.uos.de/normaliz.
   • Mihai Epure - A Schreier domain type condition in the ideal systems setting. Trimis spre publicare.
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 pregatire.

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Prezentari conferinte/Diseminarea rezultatelor

Data	Titlul prezentarii - autor
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), Moscova, Rusia
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
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