Let $P(x,y)$ be a binary cubic polynomial with integer coefficient. Let $n$ be an integer. Suppose the (complex) curve $P(x,y)=n$ is nonsingular, so is an elliptic curve. Is there any bound on the number of the integer points on it that is uniform in $n$? I'm particularly interested in the case when $P(x,y)=xy(x+y)$ (and $n$ is a nonzero integer).

$\begingroup$ There should be a bound of shape O(3^{\omega(n)}), but O(1) is doubtful. For example take an n for which there are infinitely many rational solutions and scale x and y. (You could require n to be cubefree of course, but still it seems unlikely. Also once n is cubefree the trivial bound O(d(n)) [d the divisor function] is at most O(3^{\omega(n)}), so I guess I haven't told you anything useful!) $\endgroup$ – alpoge Feb 10 '16 at 2:43

$\begingroup$ If this is of any use, the number of integer solutions to $xy(x+y) = n$ is certainly bounded above by an exponential function of $\omega(n)$, the number of distinct prime factors of $n$. Beyond this such questions are wide open, related to the problem of boundedness of ranks in families of elliptic curves. $\endgroup$ – Vesselin Dimitrov Feb 10 '16 at 2:43

$\begingroup$ @alpoge: We posted our comments at the same time (regarding exponential boundedness in $\omega(n)$)  I think you beat me by one second. I am not seeing what you mean about scaling $x$ and $y$ though  wouldn't that change $n$? $\endgroup$ – Vesselin Dimitrov Feb 10 '16 at 2:54

1$\begingroup$ @alpoge: Ah, I see, you just clear a common denominator for arbitrarily many solutions. Sorry about this :) . $\endgroup$ – Vesselin Dimitrov Feb 10 '16 at 3:03

1$\begingroup$ @VesselinDimitrov Yep absolutely! I didn't actually produce one with positive rank but I'm sure a quick computer search will. (Or else the O(1) is true by Mazur! But I'd doubt it...) $\endgroup$ – alpoge Feb 10 '16 at 3:07
As the other answers have made clear, there is a big difference if $n$ is arbitrary, or if $n$ is cubefree. But actually, the right formulation in the latter case is to instead ask for solutions $(x,y)$ satisfying $\gcd(x,y)=1$. So for any $n\in\mathbb Z\setminus\{0\}$, let $$ N^*(F,n) = \#\bigl\{(x,y)\in\mathbb Z^2 : F(x,y)=n~\text{and}~\gcd(x,y)=1\bigr\}. $$ Also let $E_{F,n}:F(x,y)=nz^3$ be the associated elliptic curve. Then there is a bound of the form [1] $$ N^*(F,n) \le C(F)^{1+\text{rank}\,E_{F,n}(\mathbb Q)}, $$ where $C(F)$ is independent of $n$. In particular, if you could find any one $F$, for example $F(x,y)=xy(x+y)$ as you specified, for which you could prove that $$ \sup_{n\in\mathbb Z\setminus\{0\}} N^*(F,n) = \infty, $$ then you would also have proven that $$ \sup_{n\in\mathbb Z\setminus\{0\}} \text{rank}\,E_{F,n}(\mathbb Q) = \infty, $$ which would be a spectacular result. I've listed a few references that deal with uniform bounds for integer points on (minimal) affine models of elliptic curves. The following MO questions/answers also seem related/relevant:
unboundedness of number of integral points on elliptic curves?
Can the number of solutions $xy(xy1)=n$ for $x,y,n \in Z[t]$ be unbounded as n varies?
[1] J.H. Silverman, Integer points and the rank of Thue elliptic curves Invent. Math. 66 (1982), 395404.
[2] D. Abramovich, Uniformity of stably integral points on elliptic curves, Invent. Math. 127 (1997), 307317.
[3] J.H. Silverman, A quantitative version of Siegel's theorem: Integral points on elliptic curves and Catalan curves, J. Reine Angew. Math. 378 (1987), 60100.
[4] M. Hindry, J.H. Silverman, The canonical height and integral points on elliptic curves, Invent. Math. 93 (1988), 419450.
Joro's linked paper is due to my thesis advisor, Professor C. L. Stewart, and in that paper he showed that if you allow nonprimitive integer pairs, you can get impressive lower bounds for Thue equations for infinitely many $n$. However, the more natural question is to ask whether one can bound the number of primitive solutions. On that, Stewart actually made a conjecture in his 1991 JAMS paper "On the number of solutions of polynomial congruences and Thue equations" which is the strongest possible: there exists an absolute constant $c$ such that for any binary form $F$ with integer coefficients, degree $d$ at least $3$, and nonzero discriminant, there exists a number $r_F$ which depends on $F$ such that for all integers $h$ with absolute value exceeding $r_F$, there are at most $c$ primitive solutions to the equation $F(x, y) = h$. This conjecture is best possible since there are always forms where $F(x, y) = 1$ has at least $d$ solutions for any degree $d$.
This conjecture has been supported in many settings. There is an arxiv preprint due to J. L. Thunder that shows that primitive solutions to Thue equations come from either an extremely good rational approximation to an algebraic number or an "exotic" lattice. He proved that the proportion of such lattices go to zero as $h$ tends to infinity.
I had thought about this conjecture for years now. To me the most convincing evidence is the work of Stoll and others on finding bounds for the number of rational points on hyperelliptic curves. The state of the art today basically shows that one can apply the ChabautyColeman method to get a bound which depends only on the degree of the form and the rank of the Jacobian. However the deficit is that this only works if the rank is smaller than the genus. This is almost surely satisfied for large degrees, since the average rank of the Jacobian of hyperelliptic curves is bounded by BhargavaGross. To prove the weak version of Stewart's conjecture where one replaces the absolute constant $c$ with a number $c(d)$ which depends on $d$, we just need to show that given a form $F$, the rank is bounded within the family of quadratic twists. This is not doable yet as far as I know.
A more encouraging result is recent work by Balakrishnan et. al who applied Chabauty to count integer points. This relaxes the condition "rank less than genus" to also allow the equality case. However, this is still not enough to prove even the weak version of Stewart's conjecture.
I would be very interested to see a resolution of his conjecture.
Here are some references:
C. L. Stewart, On the number of solutions of polynomial congruences and Thue equations, Journal of the American Mathematical Society, (4) 4 (1991), 793835. (http://www.ams.org/journals/jams/19910404/S0894034719911119199X/S0894034719911119199X.pdf)
J. L. Thunder, Thue equations and lattices, arXiv:1505.00197 [math.NT] (http://arxiv.org/abs/1505.00197)
J. S. Balakrishnan, A. Besser, J.S. Muller, Quadratic Chabauty: padic heights and integral points on hyperelliptic curves, to appear in Journal für die reine und angewandte Mathematik (http://www.degruyter.com/view/j/crelle.aheadofprint/crelle20140048/crelle20140048.xml)
Here is lower bound, via alpoge's
construction in comments:
Abstract: We shall prove that if $F$ is a cubic binary form with integer coefficients and nonzero discriminant then there is a positive number $c$, which depends on $F$, such that the Thue equation $F (x, y) = m$ has at least $c(\log{m})^{1/2}$ solutions in integers $x$ and $y$ for infinitely many positive integers $m$.