Temperedness of measures defined by polynomial equations over local fields

Abstract

We investigate the asymptotic growth of the canonical measures on the fibers of morphisms between vector spaces over local fields of arbitrary characteristic. For non-archimedean local fields we use a version of the {\L}ojasiewicz inequality (\cite{lojasiewicz1959}, \cite{hormander1958division}) which follows from Greenberg \cite{greenberg1966rational}, \cite{bollaerts1990estimate}, together with the theory of the Brauer group of local fields to construct definite forms of arbitrarily high degree, and to transfer questions at infinity to questions near the origin. We then use these to generalize results of H{\"o}rmander \cite{hormander1958division} on estimating the growth of polynomials at infinity in terms of the distance to their zero loci. Specifically, when a fiber corresponds to a non-critical value which is stable, i.e. remains non-critical under small perturbations, we show that the canonical measure on the fiber is tempered, which generalizes results of Igusa and Raghavan \cite{igusa1978lectures}, and Virtanen and Weisbart \cite{virtanen2014elementary}.