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*}
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asked Nov 29, 2019 at 1:44
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$\begingroup$ Is $Q_1$ definite or indefinite? (For $Q_2$ it's less of an issue, because the question for $Q_2 - c Q_1$ is equivalent to the one for $Q_2$ and might have a different signature.) $\endgroup$– Noam D. ElkiesCommented Nov 29, 2019 at 1:53
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2$\begingroup$ (Also, the title seems quite different from the actual question, which seems to be more like "how often do the values of one quadratic form divide the values of another?") $\endgroup$– LSpiceCommented Nov 29, 2019 at 2:36
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3$\begingroup$ Why starting with $4$ variables? What can be said about e.g. quadratic forms of $2$ variables? $\endgroup$– WhatsUpCommented Nov 29, 2019 at 2:42
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1$\begingroup$ @LSpice I liked your edit, and I made a further refinement. $\endgroup$– Stanley Yao XiaoCommented Nov 29, 2019 at 22:48
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1$\begingroup$ In the $Q_1$ definite case, it seems that $Q_2 - c Q_1 =0 $ has solutions for only finitely many $c$, and we can solve one $c$ at a time, which is just a single quadratic form in four variables. I think estimating the number of solutions by the circle method is a classically solved problem. So in fact we may assume $Q_1$ indefinite. I think the number of solutions to the previous problem provides a lower bound, but should maybe be off by a log factor because of the sum over $c$ (just thinking heuristically). $\endgroup$– Will SawinCommented Nov 29, 2019 at 23:23
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