Regularization of p-adic string amplitudes, and multivariate local zeta functions

We prove that the p-adic Koba–Nielsen-type string amplitudes are bona fide integrals. We attach to these amplitudes Igusa-type integrals depending on several complex parameters and show that these integrals admit meromorphic continuations as rational functions. Then, we use these functions to regularize the Koba–Nielsen amplitudes. As far as we know, there is no similar result for the Archimedean Koba–Nielsen amplitudes. We also discuss the existence of divergencies and the connections with multivariate Igusa’s local zeta functions.