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In this thesis we study a number of invariants of singularities of a var iety, belonging to different fields of mathematics, and we investigate c ertain conjectural connections between them.A first such invaria nt of number theoretical nature is Igusa's p-adic local zeta function Z_ f, associated to a polynomial f in several variables over the p-adic int egers. Introduced by Weil and first studied by Igusa, this holomorphic f unction (on the right-hand side of the complex plane) is very closely related to the numbers N_l of solutions of the polynomial congruences f(x) =0 mod p n->C at a point x on its special fiber {f=0}. It concerns a certain diff eomorphism of the Milnor fiber of f at x obtained by lifting a path arou nd the origin in the complex plane with respect to the Milnor fibration. This diffeomorphism gives rise to linear transformations of the singula r cohomology vector spaces of the Milnor fiber. The corresponding eigenvalues are called 'eigenvalues of monodromy' and form the second ingredie nt of the Monodromy Conjecture. Related is the Bernstein-Sato polynomial b_f of a polynomial f over the complex numbers. This invariant is defin ed in terms of differential operators, and it is well-known that every r oot of b_f induces an eigenvalue of monodromy.For a polynomial f in Z[x_1,...,x_n], the Monodromy Conjecture of Igusa predicts that ever y pole of the p-adic zeta function of f induces an eigenvalue of monodro my of f at some point of the complex zero locus of f. A stronger variant of this conjecture even expects the real part of every such pole to be a root of the Bernstein-Sato polynomial b_f of f. Denef and Loeser formulated versions of these conjectures for the topological and the motivic zeta function. During the research of the last few decades, the three zeta functions and the notions of local monodromy and Bernstein-Sato polynomials have been generalized in many ways. For example, nowadays one also associates them to several polynomials. The two conjectures we mentioned can still be stated in many of those generalized contexts. These conjectures have been verified in numerous special cases, including t he dimension two case, but up to now, no one has a clue how to prove the m in general.In this thesis we prove a new special case of the M onodromy Conjecture; more precisely, we prove the conjecture for Igusa's and the motivic zeta function of a polynomial in three variables that i s non-degenerated with respect to its Newton polyhedron.Followin g ideas of Veys, we also study a generalization of the stronger conjectu re on Bernstein-Sato polynomials, involving zeta functions associated to several polynomials and more general differential forms. Given an ideal I of C[x_1,...,x_n], we investigate whether it is possible to find a co llection G of polynomials g in C[x_1,...,x_n], such that, for all g in G , every pole of the topological zeta function of I and gdx is a root ofthe Bernstein-Sato polynomial of I, and such that all roots are realized in this way. Despite the positive results that have been obtained for the analogous question for the Monodromy Conjecture, we find a negative answer to the previous question, providing counterexamples for monomial and principal ideals in dimension two.

academic collection --- 511.6 <043> --- 512.75 <043> --- Algebraic number fields--dissertaties --- Arithmetic problems of algebraic varieties. Rationality questions. Zeta-functions--dissertaties --- Theses