Multiparametric exponential sums associated with quasi-homogeneous polynomial mappings

Abstract

We obtain sharp estimates for p-adic oscillatory integrals of the form E A ( z , f ) = ∫ A ψ ( ∑ j = 1 l z j f j ( x ) ) | d x | , where ψ is a nontrivial additive character on a non-archimedean local field K of arbitrary characteristic, and f = ( f 1 , … , f l ) : A → K l is a quasi-homogeneous polynomial mapping defined on a compact subset A ⊆ K n . We prove that if l ⩽ n , then E A ( z , f ) = O ( ‖ z ‖ K − α ) , α > 0 , as ‖ z ‖ K → ∞ , and give an explicit expression for α. If l = 1 , our estimation agrees with the one obtained by using Igusa's theory. If A = R K n , where R K is the ring of integers of K, and each f j has coefficients in R K , then E A ( z , f ) becomes a Gaussian sum depending on several parameters. The estimation of this type of oscillatory integrals occurs in the circle method and in some p-adic quantum models.