Let $Q_1, Q_2$ be two quadratic forms with integer coefficients in 4 variables $x_1, x_2, x_3, x_4$, both non-singular and not proportional. For a positive number $X$, which we may assume to be large compared to the heights of the $Q_i$’s, how does one estimate the size of the set of all $(x_1, x_2, x_3, x_4) \in \mathbb{Z}^4$ for which \begin{gather*} \max\{\lvert x_1\rvert, \lvert x_2\rvert, \lvert x_3\rvert, \lvert x_4\rvert\} \leq X \\ \text{and} \\ Q_1(x_1, x_2, x_3, x_4) \mid Q_2(x_1, x_2, x_3, x_4)\}? \end{gather*}

Let $Q_1, Q_2$ be two quadratic forms with integer coefficients in 4 variables $x_1, x_2, x_3, x_4$, both non-singular and not proportional. For a positive number $X$, which we may assume to be large compared to the heights of the $Q_i$’s, how does one estimate the size of the set of all $(x_1, x_2, x_3, x_4) \in \mathbb{Z}^4$ for which \begin{gather*} \max\{\lvert x_1\rvert, \lvert x_2\rvert, \lvert x_3\rvert, \lvert x_4\rvert\} \leq X \\ \text{and} \\ Q_1(x_1, x_2, x_3, x_4) \text{ divides } Q_2(x_1, x_2, x_3, x_4)? \end{gather*}

Let $Q_1, Q_2$ be two quadratic forms with integer coefficients in 4 variables $x_1, x_2, x_3, x_4$, both non-singular and not proportional. For a positive number $X$, which we may assume to be large compared to the heights of the $Q_i$’s, how does one estimate the size of the set

$$\displaystyle \{(x_1, x_2, x_3, x_4) \in \mathbb{Z}^4 : \max\{|x_1|, |x_2|, |x_3|, |x_4|\} \leq X, Q_1(x_1, x_2, x_3, x_4) | Q_2(x_1, x_2, x_3, x_4)\}?$$

Let $Q_1, Q_2$ be two quadratic forms with integer coefficients in 4 variables $x_1, x_2, x_3, x_4$, both non-singular and not proportional. For a positive number $X$, which we may assume to be large compared to the heights of the $Q_i$’s, how does one estimate the size of the set of all $(x_1, x_2, x_3, x_4) \in \mathbb{Z}^4$ for which \begin{gather*} \max\{\lvert x_1\rvert, \lvert x_2\rvert, \lvert x_3\rvert, \lvert x_4\rvert\} \leq X \\ \text{and} \\ Q_1(x_1, x_2, x_3, x_4) \mid Q_2(x_1, x_2, x_3, x_4)\}? \end{gather*}