... z �= 0. Indeed this condition excludes polynomials like p(z) = zd 1 or p(z) = (z1 1 +. . .+z2 n) d 2 . It is implicit in earlier works on the Hardy-Littlewood method of exponential sums and shown in =-=[M1]-=-, that if n > (d − 1)2d and p is non-singular, then there is an infinite arithmetic progression Λ, such that for each λ ∈ Λ one has Np,λ � λ n d −1 . We’ll refer to such a set Λ as a set of regular va...