Irreductibility, Factorizations, Krull Dimension, and their Computational Aspects in

Polynomials, Rings, Modules, Latices, and Grothendieck Categories

(grant ID-PCE no. 443/19.12.2008, code 1190)

Short description of the research project

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.


[AIoTe 04] T. ALBU, M. IOSIF, M.L. TEPLY, Modular QFD lattices with applications to Grothendieck categories and torsion theories, J. Algebra Appl. 3 (2004), 391-410.

[AIoTe 05] T. ALBU, M. IOSIF, M.L. TEPLY, Dual Krull dimension and quotient finite dimensionality, J. Algebra 284 (2005) , 52-79.

[AKrTe 00] T. ALBU, G. KRAUSE, M.L. TEPLY, The nilpotence of the tau-closed prime radical in rings with tau-Krull dimension, J. Algebra 229 (2000), 498-513.

[AKrTe 01] T. ALBU, G. KRAUSE, M.L. TEPLY, Bijective relative Gabriel correspondence over rings with torsion theoretic Krull dimension, J. Algebra 243 (2001), 644-674.

[AlBoBoZa 06] E. ALKAN, A.I. BONCIOCAT, N.C. BONCIOCAT, A. ZAHARESCU, A square-free criterion for polynomials using no derivatives, Proc. Amer. Math. Soc. 135 (2007), no. 3, 677-687.

[AlFi 04] M. ALLEN, M. FILASETA, A generalization of a third irreducibility theorem of I. Schur, Acta Arith.114 (2004), 183-197.

[AlFi 03] M. ALLEN, M. FILASETA, A generalization of a second irreducibility theorem of I. Schur, Acta Arith.109 (2003), 65-79.

[ANa 71] T. ALBU, C. NASTASESCU, Decomposition primaire dans les modules de torsion, C. R. Acad. Sci. Paris 273, Serie A (1971), 696-699.

[ANa 76a] T. ALBU, C. NASTASESCU, Decompositions primaires dans les categories de Grothendieck commutatives (I), J. Reine Angew. Math. 280 (1976), 172-194.

[ANa 76b] T. ALBU, C. NASTASESCU, Decompositions primaires dans les categories de Grothendieck commutatives (II), J. Reine Angew. Math. 282 (1976), 172-185.

[ANa 84] T. ALBU, C. NASTASESCU, "Relative Finiteness in Module Theory", Marcel Dekker, Inc., New York and Basel, 1984.

[AR02] M. AYAD, P. RYCKELYNCK, Algebraic independency of factors of generalized difference polynomials, cahiers du L.M.P.A février 2002, vol.167.

[ARi 01] T. ALBU, S.T. RIZVI, Chain conditions on quotient finite dimensional modules, Comm. Algebra, 29 (2001), 1909-1928.

[ASm 95] T. ALBU, P.F. SMITH, Dual relative Krull dimension of modules over commutative rings, in "Abelian Groups and Modules", edited by A. Facchini and C. Menini, Kluwer Academic Publisher, Dordrecht, pp. 1-15 (1995).

[ASm 96] T. ALBU, P.F. SMITH, Localization of Modular Lattices, Krull Dimension, and the Hopkins-Levitzki Theorem (I). Math. Proc. Cambridge Philos. Soc. 120 (1996), 87-101.

[ASm 97] T. ALBU, P.F. SMITH, Localization of Modular Lattices, Krull Dimension and the Hopkins-Levitzki Theorem (II), Comm. Algebra 25 (1997), 1111-1128.

[ASm 99] T. ALBU, P.F. SMITH, Dual Krull dimension and duality, Rocky Mountain J. Math. 29 (1999), 1153-1165.

[ASm 00] T. ALBU, P.F. SMITH, Global Krull dimension, in "Ring Theory and Representations of Algebras", edited by M. Saorin and F. Van Oystaeyen, Marcel Dekker, Inc., New York Basel, pp. 1-23 (2000).

[ATe 00] T. ALBU, M.L. TEPLY, Generalized deviation of posets and modular lattices, Discrete Math. 214 (2000), 1-19.

[AVa 98], T. ALBU, P. VAMOS, Global Krull dimension and global dual Krull dimension of valuation rings, in "Abelian Groups, Module Theory, and Topology", edited by D. Dikranjan and L. Salce, Marcel Dekker, Inc., New York, pp. 37-54 (1998).

[AnWi] D.D. ANDERSON and M. WINDERS, “Idealization of a module”, Rocky Mountain J. Math., to appear.

[Bo 89] N. BOURBAKI, “Commutative Algebra”, Chapters 1-7, Springer-Verlag, 1989.

[BL06] C. BALLOT, F. LUCA, Prime factors of A^{f(n)}-1 with an irreducible polynomial f(X), New York J. Math, 12, 39-45 (2006).

[BMV07] P. BORWEIN, M.J. .MOSSINGHOFF, J.D. VAALER., Generalizations of Goncalves' inequality, Proc. Amer. Math. Soc. 135, No. 1, 253-261 (2007).

[Bon 04] N.C. BONCIOCAT, Upper bounds for the number of factors for a class of polynomials with rational coefficients, Acta Arith. 113 (2) (2004), 175-187.

[Bon 07] N.C. BONCIOCAT, Irreducibility criteria for pairs of polynomials whose resultant is a prime number, submitted.

[BoBo 06a] N.C. BONCIOCAT, A.I. BONCIOCAT, Some classes of irreducible polynomials, Acta Arith. 123 (2006), 349-360.

[BoBo 06b] N.C. BONCIOCAT, A.I. BONCIOCAT, The irreducibility of polynomials that have one large coefficient and take a prime value, to appear in Canad. Math. Bull.

[BoBo 06c] A.I. BONCIOCAT, N.C. BONCIOCAT, A Capelli type theorem for multiplicative convolutions of polynomials, to appear in Math. Nachr.

[BoBoZa 06] N.C. BONCIOCAT, A.I. BONCIOCAT, A. ZAHARESCU, Bounds for the multiplicities of the roots for some classes of complex polynomials, Math. Inequal. Appl. 9 (1) (2006), 11-22.

[BoBoZa 07] N.C. BONCIOCAT, A.I. BONCIOCAT, A. ZAHARESCU, On the number of factors of convolutions of polynomials with integer coefficients, to appear in Rocky Mountain J. Math. vol 38 no. 2 (2008).

[Cav 00] M. CAVACHI, On a special case of Hilbert's irreducibility Theorem, J. Number Theory 82 (2000), 96-99.

[CaVaZa 02] M. CAVACHI, M. VAJAITU, A. ZAHARESCU, A class of irreducible polynomials, J. Ramanujan Math. Soc. 17 (2002), 161-172.

[Fa 99] C. FAITH, Quotient finite dimensional modules with ACC on subdirectly irreducible submodules are Noetherian, Comm. Algebra 27 (1999), 1807-1810.

[FiDo 05] M. FIILASETA, M. DOUGLAS, Irreducibility testing of lacunary 0,1-polynomials, J. Algorithms 55 (2005), 21-28.

[FiSc 04] M. FIILASETA, A. SSHINZEL, On testing the irreducibility of lacunary polynomials by cyclotomic polynomials, Math. Comp. 73 246 (2004), 957-965.

[FiLa 02] M. FIILASETA, T.-Y. LAM, On the irreducibility of the generalized Laguerre polynomials, Acta Arith. 105 (2002), no. 2, 177-182.

[FiTr 02] M.FIILASETA, O.TRIFONOV, The irreducibility of the Bessel polynomials, J.Reine Angew. Math.550 (2002), 125-140.

[Fo 67] J. FORT, Sommes directes de sous-modules coirreductibles d'un module, Math. Z. 103 (1967), 363-388.

[Fu 50] L. FUCHS, On primal ideals, Proc. Amer. Math. Soc. 1 (1950), 1-6.

[FuHeOl 04] L. FUCHS, W. HEINZER, B. OLBERDING, Commutative ideal theory without finiteness conditions: primal ideals}, Trans. Amer. Math. Soc. 357 (2004), 2771-2798.

[FuHeOl 06] L. FUCHS, W. HEINZER, B. OLBERDING, Commutative ideal theory without finiteness conditions: completely irreducible ideals, Trans. Amer. Math. Soc. 358 (2006), 3113-3131.

[Gao01] S. GAO, Absolute irreducibility of polynomials via Newton polytopes, J. Algebra, 237, 501-520 (2001).

[Gao03] S. GAO, V. RODRIGUES, Irreducibility of polynomials modulo p via Newton polytopes, J. Number Theory, 101, 32-47 (2003).

[Ha04] L. HAJDU, Irreducible polynomials in arithmetic progressions and a problem of Szegedy, Publ. Math. Debrecen, 65, 363-370 (2004).

[HeOl 05] W. HEINZER, B. OLBERDING, Unique irredundant intersections of completely irreducible ideals, J. Algebra 287 (2005), 432-448.

[Kn03} D. E. KNUTH, The Art of Computer Programming, vol. 2 Seminumerical Algorithms, Addison-Willey (2003).

[Ko05] F.KOYUNCU, Irreducible polynomials by the polytope method over integral domains, Selcuk J. Appl..Math, 6, 3-7 (2005).

[NaPo 66] C. NASTASESCU, N. POPESCU, Sur la structure des objets de certains categories abeliennes, C. R. Acad. Sci. Paris 262, Serie A (1966), 1295-1297.

[P00] L. PANAITOPOL, Minorations pour les measures de Mahler de Certains polynomes particuliers, J. Theor. Nombres Bordeaux, 12, 127-132 (2000).

[PaSt 87] L. PANAITOPOL, D. STEFANESCU, A resultant condition for the irreducibility of the polynomials, J. Number Theory 25 (1987), no. 1, 107-111.

[PaSt 88] L. PANAITOPOL, D. STEFANESCU, Factorization of the Schoenemann polynomials, Bull. Math. Soc. Sci. Math. R. S. Roumanie (N.S.) 32(80) (1988), no. 3, 259-262.

[PaSt 90] L. PANAITOPOL, D. STEFANESCU, On the generalized difference polynomials, Pacific Journal of Mathematics 143, 341--348 (1990).

[PaSt 06] L. PANAITOPOL, D. STEFANESCU, Polynomial factorizations, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 49 (97) (2006), no. 1, 69-74.

[Po 71] N. POPESCU, "Categorii Abeliene", Editura Academiei, Bucuresti, 1971.

[Ra 81] N. RADU, "Lectii de Algebra: Descompunerea Primara in Inele Comutative", Tipografia Universitatii BucurestI 1981.

[RaMu 02] M. RAM MURTY, Prime numbers and irreducible polynomials, Amer. Math. Monthly 109 (2002) no. 5, 452-458.

[SMY08] C. J. SMYTH: The Mahler measure of Algebraic Numbers. A Survey, arXIV:math/0701397v2 [Jan2008].

[St 75] B. STENSTROM, "Rings of Quotients'', Springer-Verlag, Berlin Heidelberg New York, 1975

Project coordinator:

Research team:

Research that was financially supported in 2009