This paper shows that a nontrivial uniform decay estimate for complete exponential sums modulo pr, determined by a polynomial map ${\bf P} = (P_1, P_2)$ follows from the existence of a ‘good P decomposition’ of ${\mathbb Z}_p^n$, a property that can be proved with geometric methods, and which was introduced in an earlier article by the present author.

A question of Igusa (from 1978) inquires about the singular behavior of the singular series, determined by a polynomial mapping ${\mathbf P}$:K$^n$→K$^m$, m[les ]n, where K is a local field of characteristic zero. This paper describes in geometric terms the singularities of the singular series for two classes of polynomial maps ${\mathbf P}$=(P$_1$, P$_2$):K$^n$→K$^2$. The main result, which makes possible this description, is a type of uniformization of ${\mathbf P}$ by finitely many monomial maps μ(${\mathbf x}$)=(${\mathbf x}$ $^$$_1$, ${\mathbf x}$$^ $$_2$), such that rank ($^$$_1$ $_$$_2$)= 2. This is proved using resolution of singularities. Using this result, nontrivial estimates of oscillatory integrals with phase λ$_1$P$_1$+ λ$_2$P$_2$ are possible. These will be described elsewhere.