For a system of forms of equal degree defining a non-singular complete intersection over the rationals, the number of integral zeros of height at most B is expected to obey an asymptotic formula of Hardy-Littlewood type, with a leading term given by a product of local densities. Iterating a multidimensional q-analogue of van der Corput differencing to arbitrary depth, rather than stopping after a single application, converts information about the dimension of the singular locus of the system into bounds on the associated exponential sums. This yields the expected asymptotic under a codimension hypothesis that grows more slowly in the degree than the classical requirement, with a genuine gain only once the degree is at least four, since only then can the differencing be iterated more than once.
Given polynomials f1,...,fr in Z[x1,...,xn] of degrees d1,...,dr, let N(f,B) count the integer points x with |x| <= B at which all r polynomials vanish simultaneously. When the polynomials cut out a …