Ireductibilitate, Factorizari, Dimensiune Krull, si Aspectele lor Computationale in
Obiectivele proiectului sunt urmatoarele 7 subiecte:
1. Descompuneri ireductibile in latici cu aplicatii la teorii de torsiune si categorii Grothendieck. Mai intai intentionam sa extindem rezultatul principal al lui Fort [Fo 67] privind caracterizarea modulelor M bogate in coireductibile cu ajutorul descompunerilor reduse ale lui 0 in orice submodule ale lui M, de la module la latici modulare superior continue. Apoi, consideram o problema similara inlocuind submodulele coireductibile cu submodule subdirect ireductibile. Ne asteptam ca laticele avand aceasta proprietate, pe care le numim latice bogate in complet coireductibile, sa fie chiar laticele atomice. Apoi vom aplica aceste rezultate obtinute de natura laticiala la categorii Grothendieck si la categorii de module inzestrate cu o teorie de torsiune ereditara.
2. Descompuneri primale, complet ireductibile, si primare in module peste inele comutative arbitrare. Dupa cum Teorema Lasker-Noether a fost extinsa de la inele Noetheriene la module Noetheriene cf. [Bo 89], obiectivul urmator este extinderea de la ideale la module peste inele comutative a unor rezultate privind idealele primale si complet ireductibile datorate lui Fuchs, Heinzer, si Olberding [FuHeOl 04], [FuHeOl 06], si Heinzer, Olberding [HeOl 05], investigarea legaturilor lor cu descompunerile primare, cat si unicitatea si iredundanta unor asemenea descompuneri.
3. Dimensiunea Krull, dimensiunea Krull duala duala si Teorema Faith.
i) Legatura intre dimensiunea Krull si dimensiunea Krull duala a unui inel. O problema deschisa formulata de Albu si Smith in 1993 intreaba daca dimensiunea Krull duala a oricarui inel (comutativ) R avand dimensiune Krull este marginita superior de dimensiunea Krull a acestuia. Rezultate partiale privind aceasta problema au fost obtinute de Albu si Smith ([ASm 95], [ASm 99]). Ne propunem sa investigam aceasta problema, precum si relativizarea ei in raport cu o teorie de torsiunii ereditara.
ii) Teorema lui Faith. Un rezultat al lui Faith [Fa 99] stabileste ca un modul drept M este Noetherian daca si numai daca M este QFD (adica quotient (Goldie) finite dimensional) si satisface conditia lanturilor ascendente pentru submodule subdirect ireductibile. Rezultatul lui Faith poate fi reformulat intr-un context de dimensiune Krull duala pentru laticea L(M) a tuturor submodulelor lui M si a submultimii sale S(M) a submodulelor subdirect ireductibile ale lui M, dupa cum urmeaza: Dimensiunea Krull duala a lui L(M) este cel mult 0 daca si numai daca L(M) este latice QFD si dimensiunea Krull duala a posetului S(M) este cel mult 0. O problema naturala, inca deschisa, formulata de Albu si Rizvi in [ARi 01] este daca reformularea de mai sus a teoremei lui Faith ramane valabila pentru un ordinal arbitrar alpha in loc de 0. Un raspuns pozitiv la aceasta intrebare pentru orice ordinal finit, este dat in Albu, Iosif, Teply [AIoTe 05]. Ne asteptam sa demonstram acest rezultat pentru un ordinal arbitrar transfinit.
4. Estimari ale inaltimilor si normelor Bombieri pentru factorii polinoamelor. Vom rafina unele inegalitati privind inaltimea si norma Bombieri a divizorilor unui polinom intr-o nedeterminata avand coeficientii numere complexe. Pentru polinoamele cu coeficienti intregi vom obtine estimari mai fine decat cele date de inegalitatile lui Landau si Beauzamy.
5. Margini pentru valorile absolute si multiplicitatile radacinilor polinoamelor. Vom obtine noi rezultate privind localizarea radacinilor unui polinom cu coeficienti reali sau complecsi. Vor fi obtinute margini superioare pentru multiplicitatile radacinilor, in particular criterii de separabilitate.
6. Aplicatii ale metodei poligonului Newton la factorizarea polinoamelor. Vom utiliza proprietati ale pantelor poligonului Newton asociat unui produs de polinoame, pentru studierea factorizarii polinoamelor in mai multe nedeterminate peste corpuri algebric inchise. In particular vor fi obtinute noi criterii de ireductibilitate.
7. Studiul ireductibilitatii combinatiilor liniare de polinoame relative prime, grupuri Cogalois si conjectura Hardy-Littlewood. Se va studia ireductibilitatea anumitor combinatii liniare de polinoame relativ prime intr-una sau mai multe nedeterminate. In particular vom obtine noi criterii de ireductibilitate pentru polinoame de tip Schonemann si vom studia grupurile Cogalois pentru diverse clase de polinoame. Vom investiga aspectele algebro-analitice si computationale ale conjecturii Hardy-Littlewood.
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Toma Albu, Cercetator Stiintific gradul I in Institutul de Matematica "Simion Stoilow" al Academiei Romane
Doru Stefanescu
Alexandru Gica
Nicolae Ciprian Bonciocat
Vlad Copil - doctorand
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