The monodromy conjecture for high order poles, and norms of residue antiderivatives

Abstract

We prove the strong local monodromy conjecture for poles of order at least $n-1$ of the naive motivic zeta-function for a complex polynomial of $n$ variables. Moreover, for every such pole $s$, we construct an explicit homology $\exp(2\pi i s)$-eigencycle in a neighborhood of any given 1-dimensional exceptional stratum contributing the pole. The proof combines topological techniques developed by the author with a fact of birational geometry discovered by AI.