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.