The zeros of a quadratic form at square-free points

Let F(x1,…,xn)F(x1,…,xn) be a nonsingular indefinite quadratic form, n=3n=3 or 4. For n=4n=4, suppose the determinant of F is a square. Results are obtained on the number of solutions ofF(x1,…,xn)=0F(x1,…,xn)=0 with x1,…,xnx1,…,xn square-free, in a large box of side P. It is convenient to count solutions with weights. LetR(F,w)=∑F(x)=0μ2(x)w(xP), where w   is infinitely differentiable with compact support and vanishes if any xi=0xi=0, whileμ2(x)=μ2(|x1|)⋯μ2(|xn|).μ2(x)=μ2(|x1|)⋯μ2(|xn|). It is assumed that F is robust in the sense thatdetM1⋯detMn≠0,detM1⋯detMn≠0, where MiMi is the matrix obtained by deleting row i and column i from the matrix M of F  . In the case n=3n=3, there is the further hypothesis that −detM1, −detM2, −detM3 are not squares. It is shown that R(F,w)R(F,w) is asymptotic toenσ∞(F,w)ρ∗(F)Pn−2logP,enσ∞(F,w)ρ∗(F)Pn−2logP, where en=1en=1 for n=4n=4, en=12 for n=3n=3. Here σ∞(F,w)σ∞(F,w) and ρ∗(F)ρ∗(F) are respectively the singular integral and the singular series associated to the problem. The method is adapted from the approach of Heath-Brown to the corresponding problem with x1,…,xnx1,…,xn unrestricted integer variables.