On singularity properties of convolutions of algebraic morphisms

Let K be a field of characteristic zero, X and Y be smooth K-varieties, and let V be a finite dimensional K-vector space. For two algebraic morphisms phi:X -> V and psi:Y -> V we define a convolution operation, phi*psi:XxY -> V, by phi*psi(x,y)=phi(x)+psi(y). We then study the singularity properties of the resulting morphism, and show that as in the case of convolution in analysis, it has improved smoothness properties. Explicitly, we show that for any morphism phi:X -> V which is dominant when restricted to each irreducible component of X, there exists N is an element of N such that for any n>N the nth convolution power phi(n):=phi*...*phi is a flat morphism with reduced geometric fibers of rational singularities (this property is abbreviated (FRS)). By a theorem of Aizenbud and Avni, for K=Q, this is equivalent to good asymptotic behavior of the size of the Z/p(k)Z-fibers of phi(n) when ranging over both p and k. More generally, we show that given a family of morphisms {phi(i):X-i -> V} of complexity D is an element of N (i.e. that the number of variables and the degrees of the polynomials defining X-i and phi(i) are bounded by D), there exists N(D)is an element of N such that for any n>N(D), the morphism phi(1)*...*phi n is (FRS).