# Density of integral points on affine cubic surfaces of a certain type

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?

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).
• 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. Jul 5, 2020 at 12:17
• 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? Jul 5, 2020 at 12:41
• 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. Jul 5, 2020 at 13:10