Multiparametric exponential sums associated with quasi-homogeneous polynomial mappings

We obtain sharp estimates for p-adic oscillatory integrals of the form where ψ is a nontrivial additive character on a non-archimedean local field K of arbitrary characteristic, and is a quasi-homogeneous polynomial mapping defined on a compact subset A Kn. We prove that if l n, then , α>0, as z K→∞, and give an explicit expression for α. If l=1, our estimation agrees with the one obtained by using Igusaʹs theory. If , where RK is the ring of integers of K, and each fj has coefficients in RK, then EA(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.