| Résume | To every polynomial (multivariate, complex coefficients) we can associate two classical singularity invariants: its Bernstein—Sato polynomial; its motivic zeta function. The former has finitely many roots, comes from the theory of algebraic differential operators, and represents solutions to a sort of universal differential equation. The latter has finitely many "poles". comes from the theory of motivic integration, and its "poles" encode subtle relations amongst the numerical data of a (any) embedded resolution. The Strong Monodromy Conjecture is wide open, even in three variables, and promises that poles of the motivic zeta function are contained in the roots of the Bernstein—Sato polynomial. I will discuss a proof of this conjecture for a class of polynomials in three variables, namely homogeneous polynomials (not necessarily reduced) whose associated reduced projective curve has only quasi-homogeneous singularities. Even outside of these special hypotheses, our results show that if the Strong Monodromy Conjecture holds, there is a surprising connection between certain linear syzygies of Jacobian ideals and vanishing of candidate poles of the motivic zeta function. Based on arXiv 2602.20922, joint with Wim Veys.
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