The circle method and pairs of quadratic forms
Let \[ \phi_1(x_1,\ldots,x_4)=a_1x_1^2+ \ldots + a_4x_4^2 \] \[ \phi_2(x_1,\ldots,x_4)=b_1x_1^2+ \ldots + b_4x_4^2 \] be quadratic forms with integer coefficients, and for \(B>0\) let \(M(B) \) be the number of vectors \(x\in{\mathbb Z}^4\) of size \(B\) such that \( \phi_1(x)=0 \) and \(\phi_2(x) \) is a non-zero square. Assume that the discriminant \(\alpha=a_1a_2a_3a_4\) is not a square, and that all the \(2\times 2\) minors of the matrix \[ \begin{pmatrix} a_1&a_2&a_3&a_4\\ b_1&b_2&b_3&b_4 \end{pmatrix} \] are non-zero. Let \({\mathcal M}\) be the product of all the \(2\times 2\) minors. The authors prove that if \(\alpha\) is not a square and \({\mathcal M}\neq 0\) then \[ M(B)\ll B^{\frac 95+\varepsilon}, \] where the implied constant depends on the quadratic forms \(\phi_i \) and on \(\varepsilon\).
scientific article; zbMATH DE number 5862108