Polynomials represent a central object of study in several important areas of Mathematics, like Commutative Algebra, Algebraic Geometry, Arithmetics and Number Theory. This research project aims to study the polynomial ideals using methods coming from three different important directions in contemporary mathematical research : 1) Combinatorial and computational aspects of commutative algebra 2) Methods of homological algebra in the study of the resolutions of polynomial ideals 3) Arithmetical, combinatorial and algebraic factorization methods in polynomial rings.The main objectivesof this research project, corresponding to each of the three research directions mentioned above, are : 1) - The computation of some algebraic invariants (Castelnuovo-Mumford regularity, depth etc.) for special classes of monomial ideals and the search of linear bounds for the Castelnuovo-Mumford regularity for the powers of Borel type ideals - The study of weak and strong Lefschetz property for complete intersection ideals - The characterization of Koszulness of the graded ring associated to the incidence algebra of a poset - The description of a class of monomial ideals whose Betti numbers are characteristic free by means of the Koszul homology modules - The study of the Koszul homology modules of Cohen-Macaulay and Gorenstein modules 2) - The use of vector bundles and derived categories techniques in the study of polynomial equations of two-codimensional subvarieties of a projective space via their general hyperplane sections - The use of this kind of techniques for some related problems concerning the extension/restriction of vector bundles on projective spaces 3) - The study of multiplicative maps on the ring of polynomials with integer coefficients with applications in factorization problems - Obtaining irreducibility and separability conditions for polynomials with integer coefficients, and for multivariate polynomials over arbitrary fields.The main activities,corresponding to the objectives mentioned above are: gathering scientific information, research visits, scientific research, editing, scientific results dissemination. - The objectives mentioned at the paragraphs 1) and 2) abovehave been completely accomplishedduring the first three stages of the project (2007, 2008 and 2009). - The objectives mentioned at paragraph 3) correspond to the final stage of the project (2010), and are about to be accomplished.

Iustin-Gh. Coanda, Senior Researcher I in the Institute of Mathematics "Simion Stoilow" of the Romanian Academy

Iustin Coanda

Dorin Popescu

Cristodor Ionescu

Nicolae Ciprian Bonciocat

Marius Vladoiu

Mircea Cimpoeas

Dumitru Stamate

Mihai Epure

- At the beginning of the project, three members of our research team (M. Cimpoeas, D. Stamate and M. Epure) were Ph. D. students. The topics of their Ph. D. thesis are compatible with the objectives of this project. Reaching these objectives implies a pluridisciplinary effort, necessitating methods of combinatorial commutative algebra, arithmetics and algebraic geometry. The scientific competence of the senior members of the team agrees with this pluridiscilinary character of the project. - M. Cimpoeas obtained his PhD degree in 2008 and D. Stamate defended his PhD thesis in July 2009. M. Epure will complete his PhD thesis in 2010. - Each of them had significant scientific contributions to the objectives of this project, as one can see by looking at the list of publications below.

- Computation of some algebraic invariants (Castelnuovo – Mumford regularity, depth and dimension) for an important class of monomial ideals, namely the mixed product ideals - Characterization of the monomial ideals of Borel type using the notion of stable monomial ideal. As an application, one gets an upper bound for the Castelnuovo – Mumford regularity of the product of two ideals of Borel type. - Results in connection with Stanley' s Conjecture concerning the Stanley depth of monomial modules: a) the proof of Stanley's Conjecture for monomial quotients of polynomial rings in at most 5 indeterminates; b) a computation algorithm of the Stanley depth of a quotient of two monomial ideals; c) estimation of Stanley depth for complete intersection monomial ideals; d) proof of Stanley' s Conjecture for 3 - generated monomial ideals and for the corresponding quotient algebra; e) estimation of Stanley depth for the powers of the homogeneous maximal ideal of a polynomial ring; f) the proof of Stanley' s Conjecture for intersections of two irreducible monomial ideals and for quotients of the polynomial ring by intersections of 3 irreducible monomial ideals. - An analysis of a Conjecture of Moreno concerning the generic initial ideal of a homogeneous complete intersection ideal of a polynomial ring, including a proof of the Conjecture for 3 – generated ideals of this type. - Introduction of a new class of algebras, namely the reduced incidence algebras, and the characterization of the right monomial ideals of these algebras with Koszul property. This unifies and extends three previously known characterizations for: a) Koszul incidence algebras (Polo, Woodcock), b) Koszul affine semigroup rings (Peeva-Reiner-Sturmfels), c) square-free monomial ideals with linear resolutions (Eagon-Reiner, Herzog-Hibi, Herzog-Reiner-Welker). - Generalization of Horrocks Correspondence for vector bundles on projective spaces to coherent sheaves, including ideal sheaves of subvarieties of projective spaces. - Characterization of infinitely stably extendable vector bundles on projective spaces and a new effective version of the Babylonian Tower Theorem. - Proof of a Conjecture of H. Brenner concerning the existence of stable monomial syzygy bundles on projective spaces. - Irreducibility and separability criteria for some classes of polynomials with integer coefficients and with coefficients in arbitrary fields. - Characterizations of some classes of quadratic real fields with principal ring of integers.

C. Ionescu, G. Rinaldo: Some algebraic invariants related to mixed product ideals, Arch. Math. 91, no. 1, 20-30 (2008)

(ISI impact factor 0.5);

M. Cimpoeas: Some remarks on Borel type ideals, Commun. Algebra 37, no. 2, 724-727 (2009) (ISI impact factor 0.337);

J. Herzog, M. Vladoiu, X. Zheng: How to compute the Stanley depth of a monomial ideal, J. Algebra 322, no. 9, 3151-3169

(2009) (ISI impact factor 0.63);

D. Popescu: Stanley depth of multigraded modules, J. Algebra 321, no. 10, 2782-2797 (2009) (ISI impact factor 0.63);

M. Cimpoeas: Stanley depth of complete intersection monomial ideals, Bull. Math. Soc. Sci. Math. Roumanie 51(99), no. 3,

205-211 (2008) (ISI expanded list);

V. Reiner, D.I. Stamate: Koszul incidence algebras, affine semigroups, and Stanley-Reisner ideals, Adv. Math. 224,

no. 6, 2312-2345 (2010) (ISI impact factor 1.403);

M. Cimpoeas: Stanley depth of monomial ideals with small number of generators, Central European Journal of Mathematics,

vol. 7, no. 4, 629-634 (2009) (ISI impact factor 0.361);

M. Cimpoeas: Some remarks on the Stanley's depth of multigraded modules, Le Matematiche, vol. LXIII, fasc. II, 165-171

(2008);

M. Cimpoeas: Lefschetz property for complete intersections, Analele Univ. Buc., Matematica, Anul LVIII, 125-144 (2009);

I. Coanda: The Horrocks correspondence for coherent sheaves on projective spaces, Homology, Homotopy Appl. 12, no. 1,

327-353 (2010) (ISI impact factor 0.609);

I. Coanda: Infinitely stably extendable vector bundles on projective spaces, Arch. Math. 94, no. 6, 539-545 (2010)

(ISI impact factor 0.373);

I. Coanda: On the stability of syzygy bundles, International J. Math. (to appear) (ISI impact factor 0.608);

A.I. Bonciocat, N.C. Bonciocat: On the irreducibility of polynomials with leading coefficient divisible by a large prime power,

Amer. Math. Monthly 116, no. 8, 743-745 (2009) (ISI impact factor 0.361);

A.I. Bonciocat, N.C. Bonciocat and A. Zaharescu: On the irreducibility of polynomials that take a prime power value,

Bull. Math. Soc. Sci. Math. Roumanie 54(102), (2011) (ISI impact factor 0.554);

D. Popescu, M.I. Qureshi: Computing the Stanley depth, J. Algebra 323, no. 10, 2943-2959 (2010) (ISI impact factor 0.632);

M. Epure, A. Gica: Principal quadratic real fields in connection with some additive problems,

Bull. Math. Soc. Sci. Math. Roumanie 53(101), no. 3, 251-259 (2010) (ISI impact factor 0.554);

N.C. Bonciocat, A. Zaharescu: Irreducible multivariate polynomials obtained from polynomials in fewer variables II.