The following is from:

C.L. Stewart, "On the number of solutions of polynomial congruences and Thue equations", Journal of the American Math. Soc., 4 (1991), 793-835.

"Bombieri and Schmidt showed (in E. Bombieri and W.M. Schmidt, "On Thue's Equation", Invent. Math, 88., (1987), 69-81) that $F(x,y) = h$ where $F(x,y) = a_r x^r + a_{r-1}x^{r-1}y + \cdots + a_0 y^r$ may have at least $r$ distinct primitive solutions (a solution $(x,y)$ is primitive if $\gcd(x,y) = 1$). They gave the example $$\displaystyle F(x,y) = x^r + a(x-y)(2x-y)\cdots(rx -y),$$ where $a$ is a non-zero integer. Then $(1,1), (1,2), \cdots, (1,r)$ are primitive solutions of $F(x,y) = 1$. We do not believe that for a fixed form $F$ there are infinitely many integers $h$ for which $F(x,y) = h$ has this many primitive solutions if $r$ is large. Indeed we conjecture that there exists an absolute constant $c_0$ such that for any binary form $F \in \mathbb{Z}[x,y]$ with non-zero discriminant and degree at least three there exists a number $C$, which depends on $F$, such that if $h$ is larger than $C$ then the Thue equation $F(x,y) = h$ has at most $c_0$ solutions in coprime integers $x$ and $y$.

I am wondering what progress, if any, there is on this conjecture.

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  • $\begingroup$ Your profile indicates that you're in Waterloo. Have you tried asking Cam Stewart directly? $\endgroup$ – Faisal Jun 6 '11 at 20:35
  • $\begingroup$ I did indeed, but thought I would ask the community at large. $\endgroup$ – Stanley Yao Xiao Jun 6 '11 at 21:16
  • $\begingroup$ To get an idea of what is happening for a fixed $F$ (if $d \ge 5$), you should apply the Bombieri-Lang conjecture to the surface with homogeneous equation $F(x,y)=F(z,w)$. $\endgroup$ – Felipe Voloch Jun 7 '11 at 12:24

Perhaps I'm getting lost in the quantifiers, but this conjecture seems to me to be inconsistent with "We prove that there are infinitely many inequivalent cubic binary forms $F$ with content $1$ for which the Thue equation $F(x,y)=m$ has $\gg(\log m)^{6/7}$ solutions in integers $x$ and $y$ for infinitely many integers $m$,'' taken from a later paper by Stewart, Integer points on cubic Thue equations, C. R. Math. Acad. Sci. Paris 347 (2009), no. 13-14, 715–718, MR2543969 (2010j:11064). Mahler already had this in 1935, with $1/4$ in place of $6/7$.

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  • $\begingroup$ I believe the novelty in the conjecture, which diverges from the result given above (and earlier results due to Mahler and Silverman), is that the integers $x,y$ are now required to be co-prime. For example one can see if $x,y$ are not co-prime then one can foresee-ably get lots of solutions to $F(x,y) = m$ if $m$ has lots of divisors that are perfect $r$ powers, where $r$ is the degree of the form. $\endgroup$ – Stanley Yao Xiao Jun 7 '11 at 1:44
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    $\begingroup$ To expand a bit on Stanley's answer, the way that one gets lots of solutions (with $\gcd(x,y)$ large) is as follows. Take an equation $F(x,y)=c$ whose set of rational solutions (Mordell-Weil group) has rank $r$, take all solutions $P_1,...,P_n$ of height less than $H$, and clear all of their denominators to get an equation $F(x,y)=cd^3$ that has $n$ integer solutions. Rough height estimates show that the number of solutions is $\gg(\log d)^{r/(r+2)}$. As Stanley says, this idea is due to Mahler. Mahler's argument easily gives exponent $1/3$, using a single point of infinite order. $\endgroup$ – Joe Silverman Jun 7 '11 at 2:52
  • $\begingroup$ This discussion reminds me of a question that I've thought about over the years, with no success. Is it true that for every $\epsilon>0$ and binary form $F(x,y)$ with no repeated factors, the number of integer solutions to $F(x,y)=m$ is $\le C_{F,\epsilon}(\log m)^{1+\epsilon}$, where, as indicated, the constant may depend on $F$ and $\epsilon$, but is independent of $m$? $\endgroup$ – Joe Silverman Jun 7 '11 at 2:56
  • $\begingroup$ OK, I missed the "coprime" part. Sorry. $\endgroup$ – Gerry Myerson Jun 7 '11 at 5:24
  • $\begingroup$ @Joe Silverman: Professor Silverman, what is the motivation for the factor of $(\log m)^{1 + \epsilon}$? Is there any heuristic as to why this is a reasonable bound? $\endgroup$ – Stanley Yao Xiao Jun 7 '11 at 18:16

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