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For a nonzero ideal I of C[x_1,...,x_n], with 0 in supp I, a generalizationof a conjecture of Igusa - Denef - Loeser predicts that every pole of itstopological zeta function is a root of its Bernstein-Sato polynomial. However,typically only a few roots are obtained this way. Following ideas of Veys, westudy the following question. Is it possible to find a collection G ofpolynomials g in C[x_1,...,x_n], such that, for all g in G, every pole of thetopological zeta function associated to I and the volume form gdx on the affinen-space, is a root of the Bernstein-Sato polynomial of I, and such that allroots are realized in this way. We obtain a negative answer to this question,providing counterexamples for monomial and principal ideals in dimension two,and give a partial positive result as well.

Zeta functions and Bernstein-Sato polynomials for ideals in dimension two