Irreductibility, Factorizations, Krull Dimension, and their Computational Aspects in
The project objectives are the following 7 topics:
1. Irreducible decompositions in lattices with applications to torsion theories and Grothendieck categories. First, we intend to extend from modules to upper continuous modular lattices the main result of Fort [Fo 67] concerning the characterization of modules M rich in coirreducibles by means of irredundant irreducible decompositions of 0 in any submodule of M. Then, we consider a similar problem by replacing coirreducible submodules with subdirectly irreducible submodules. We expect that the lattices having this property, we call lattices rich in completely coirreducibles, are exactly the atomic lattices. Then, we will apply these lattice-theoretical results obtained to Grothendieck categories and module categories equipped with a hereditary torsion theory.
2. Primal, completely irreducible, and primary decompositions in modules over arbitrary commutative rings. As the Lasker-Noether Theorem has been extended from Noetherian rings to Noetherian modules cf. [Bo 89], a next goal is to extend from ideals to modules over commutative rings some results of L. Fuchs, W. Heinzer, B. Olberding [FuHeOl 04], [FuHeOl 06], and W. Heinzer, B. Olberding [HeOl 05] concerning primal and completely irreducible ideals, to investigate their links with primary decompositions, as well as the uniqueness and irredundancy of such decompositions.
3. Krull dimension, dual Krull dimension, and the Faith’ s theorem.
(i) The relationship between Krull dimension and dual Krull dimension of a ring. An open question raised by Albu and Smith in 1993 asks whether the dual Krull dimension of any (commutative) ring R having Krull dimension is bounded above by its Krull dimension. Partial results concerning this question have been obtained by Albu and Smith ([ASm 95], [ASm 99]). We intend to deal with this problem as well as with its relativization with respect to a hereditary torsion theory.
(ii) Faith's Theorem. A nice result of C. Faith [Fa 99] establishes that a right module M is Noetherian if and only if M is QFD (i.e., quotient (Goldie) finite dimensional) and satisfies the ascending chain condition on subdirectly irreducible submodules. Faith's result can be reformulated in a dual Krull dimension setting for the lattice L(M) of all submodules of M and its subset S(M) of subdirectly irreducible submodules of M as follows: the dual Krull dimension of L(M) is at most 0 if and only if L(M) is a QFD lattice and the dual Krull dimension of the poset S(M) is at most 0. A natural, still open, question raised by Albu and Rizvi in [ARi 01] asks whether the above reformulation of Faith's result holds for an arbitrary ordinal alpha instead of 0. A positive answer to this question for any finite ordinal in given in Albu, Iosif, Teply [AIoTe 05]. We expect to prove the result for an arbitrary transfinite ordinal.
4. Heights and Bombieri’s norms estimates for polynomials factors. We will refine some inequalities concerning the height and the Bombieri’s norm of the divisors of a univariate complex polynomial. For polynomials with integer coefficients we will obtain sharper estimates than those given by the inequalities of Landau and Beauzamy.
5. Bounds for the absolute values and the multiplicities of the polynomial roots. We will obtain new results on the root location for real and complex polynomials. We will obtain upper bounds for the multiplicities of the roots, in particular separability criteria.
6. Applications of Newton’s polygon method to polynomial factorization. We will use the properties of the slopes in the Newton polygon associated to a product of polynomials, in the study of the factorization of multivariate polynomials over an algebraically closed field. In particular, some new irreducibility criteria will be obtained.
7. The study of the irreducibility of linear combinations of relatively prime polynomials, Cogalois groups and the Hardy-Littlewood conjecture.
We will study the irreducibility of some classes of univariate and multivariate polynomials that can be expressed as linear combinations of relatively prime polynomials. In particular we will obtain new irreducibility conditions for Schoenemann–like polynomials, we will study the Cogalois groups for various classes of polynomials. We will also investigate some algebraic - analytical and computational aspects of the Hardy-Littlewood conjecture.
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Albu Toma, Senior Researcher I in the Institute of Mathematics "Simion Stoilow" of the Romanian Academy
Doru Stefanescu
Alexandru Gica
Nicolae Ciprian Bonciocat
Vlad Copil – Ph. D student
T. Albu & J. Van Den Berg, "An indecomposable non-locally finitely generated Grothendieck category with simple objects, J. Algebra 321(2009), 1358-1545.
T. Albu, "Completely irreducible meet decompositions in lattices, with applications to Grothendieck categories and torsion theories (I)", Bull. Math. Soc. Sci. Math. Roumanie 52(100) (2009), 393-419.
T. Albu, "Completely irreducible meet decompositions in lattices, with applications to Grothendieck categories and torsion theories (II)", Bull. Math. Soc. Sci. Math. Roumanie 53(101) (2010), to appear.
D. Stefanescu: "Inequalities on real roots of polynomials", to appear in MIA (Mathematical Inequalities and Applications), no. 4/2009.