Submitted by ANA KARLA PEREIRA RODRIGUES (anakarla_@hotmail.com) on 2017-08-28T14:05:33Z No. of bitstreams: 1 arquivototal.pdf: 917638 bytes, checksum: 80f3b489c75001e1f83c7c60e54c0c74 (MD5) / Approved for entry into archive by Viviane Lima da Cunha (viviane@biblioteca.ufpb.br) on 2017-08-28T15:55:51Z (GMT) No. of bitstreams: 1 arquivototal.pdf: 917638 bytes, checksum: 80f3b489c75001e1f83c7c60e54c0c74 (MD5) / Made available in DSpace on 2017-08-28T15:55:51Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 917638 bytes, checksum: 80f3b489c75001e1f83c7c60e54c0c74 (MD5) Previous issue date: 2015-07-16 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / The maingoalofthisdissertationisthepresentationofconcepts,examplesand characterizations{bothclassicalandrecent{concerningtheimportantandin uential theory oftheso-called freedivisors in thestandardhomogeneouscase.Tothisend,we beginwithabasicstudyonderivationsandwefocusonthemoduledubbed tangential idealizer of agivenhomogeneouspolynomial,whichgeometricallycorrespondstothe moduleoflogarithmicvector eldsalongthegivenprojectivehypersurface(thedivisor is saidtobe free if suchmoduleisfreeoverthegradedpolynomialring).Wewillalso discuss, inparticular,resultsaboutfreedivisorsintheprojectiveplane. / O principalobjetivodestadisserta c~ao eaapresenta c~aodeconceitos,exemplose caracteriza c~oes{tantocl assicasquantorecentes{arespeitodaimportanteein uente teoria doschamados divisoreslivres no casohomog^eneopadr~ao.Paraesta nalidade, iniciamos comumestudob asicosobrederiva c~oesefocalizamosnom odulodenomi- nado idealizadortangencial de umdadopolin^omiohomog^eneo,oquegeometricamente correspondeaom odulodoscamposvetoriaislogar tmicosaolongodahipersuperf cie projetivadada(odivisor edito livre quando talm odulo elivresobreoanelgraduado de polin^omios).Tamb emdiscutiremos,emparticular,resultadossobredivisoreslivres no planoprojetivo.

We study a multivariate version of the Bernstein–Sato polynomial, the so-called Bernstein–Sato ideal, associated to an arbitrary factorization of an analytic germ <i>f - f</i>₁···<i>f</i>_r. We identify a large class of geometrically characterized germs so that the <i>D</i>_X,x[<i>s</i>₁,...,<i>s</i>_r]-annihilator of <i>f</i>^s₁¹···<i>f</i>^s_r^r admits the simplest possible description and, more-over, has a particularly nice associated graded object. As a consequence we are able to verify Budur’s Topological Multivariable Strong Monodromy Conjecture for arbitrary factorizations of tame hyperplane arrangements by showing the zero locus of the associated Bernstein–Sato ideal contains a special hyperplane. By developing ideas of Maisonobe and Narvaez-Macarro, we are able to find many more hyperplanes contained in the zero locus of this Bernstein–Sato ideal. As an example, for reduced, tame hyperplane arrangements we prove the roots of the Bernstein–Sato polynomial contained in [−1,0) are combinatorially determined; for reduced, free hyperplane arrangements we prove the roots of the Bernstein–Sato polynomial are all combinatorially determined. Finally, outside the hyperplane arrangement setting, we prove many results about a certain <i>D</i>_X,x-map ∇_<i>A</i> that is expected to characterize the roots of the Bernstein–Sato ideal.