(grant PNII-IDEI-PCE no. 51/28.09.2007, CNCSIS code 304/2007)

Short description of the research project

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 objectives of 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 

   - 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 

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) above
have been completely accomplished during 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.

Project coordinator

Research team from the Institute of Mathematics ``Simion Stoilow'' of the Romanian Academy

Young researchers and their contribution to the project

   - 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.

The main scientific results of the project

    - 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.

Research that was financially supported in 2007

Research that was financially supported in 2008

Research that was financially supported in 2009

Research that was financially supported in 2010