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Let $Q(x,y,z)$ be a cubic polynomial with integer coefficients, such that the terms $x^3, y^3, z^3$ do not appear. That is, it is at most quadratic in each of the variables $x,y,z$.

Is there a general method to count integral points $(a,b,c)$ with $\max\{|a|, |b|, |c|\} \leq T$ on the affine cubic surface defined by $Q(x,y,z) = 0$?

The prototypical example is the Markoff surface defined by $Q(x,y,z) = x^2 + y^2 + z^2 - xyz$. Here Zagier showed that the density of integral points is asymptotic to $C (\log T)^2$ for some explicit constant $C > 0$. This is generalized by Baragar and Umeda in this paper.

Their method depends on an explicit descent on the Markoff surface, namely that for a given point $(a,b,c)$ the point $(bc - a, b, c)$ is also a point. Zagier then showed that all integral solutions are generated from the fundamental solution $(3,3,3)$ and by permuting the variables and applying the above map. This means that the set of solutions grows exponentially in size, hence giving the bound $O((\log T)^2)$.

Is there something similar that can be done, using only the fact that all of the monomials in $Q$ are at most quadratic in each variable?

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Not in general.

The involutions of the Markov surface such as $(a,b,c) \leftrightarrow (bc-a,b,c)$ preserve integral points because $x^2 + y^2 + z^2 - xyz$ is a monic quadratic polynomial in each variable. That works more generally for any polynomial of the form $Q(x,y,z) = x^2 + y^2 + z^2 - L(x,y,z)$ with $L$ linear in each variable, though already in this special case $Q=0$ can have infinitely many solutions (it doesn't even have to be irreducible, e.g. $(x \pm y \pm z)^2$ works). For the general cubic polynomial with no $x^3,y^3,z^3$ terms, there are still rational involutions that fix three of the variables, but they have denominators and thus need not send integral points to integral points (and occasionally will send a rational or even integral point to infinity, when the quadratic coefficient w.r.t. one of the variables vanishes).

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  • $\begingroup$ Thank you for your answer. Are there ways to detect when there are "many" fundamental solutions to $Q = k$? Baragar and Umeda gave some examples, but I don't see how to think about this in the general case. $\endgroup$ Jul 5, 2020 at 12:17
  • $\begingroup$ You're welcome. For $Q=k$, do you mean thie for the specific case of $Q = x^2+y^2+z^2-xyz$, or any $Q$ satisfying the conditions of your question? $\endgroup$ Jul 5, 2020 at 12:41
  • $\begingroup$ I would like as general a statement as possible; if that's too much to ask then just understanding the case $Q = x^2 + y^2 + z^2 - xyz$ would be great since that will likely help me get a "feeling" for this type of phenomenon. $\endgroup$ Jul 5, 2020 at 13:10

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