Multiparametric exponential sums associated with quasi-homogeneous polynomial mappings

We obtain sharp estimates for p-adic oscillatory integrals of the formEA(z,f)=∫Aψ(∑j=1lzjfj(x))|dx|, where ψ is a nontrivial additive character on a non-archimedean local field K   of arbitrary characteristic, and f=(f1,…,fl):A→Kl is a quasi-homogeneous polynomial mapping defined on a compact subset A⊆KnA⊆Kn. We prove that if l⩽nl⩽n, then EA(z,f)=O(‖z‖K−α), α>0α>0, as ‖z‖K→∞‖z‖K→∞, and give an explicit expression for α  . If l=1l=1, our estimation agrees with the one obtained by using Igusa's theory. If A=RKn, where RKRK is the ring of integers of K  , and each fjfj has coefficients in RKRK, then EA(z,f)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.