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?


1 Answer 1


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).

  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.