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The classical Thue equation is $$\displaystyle F(x,y) = h,$$ for a binary form $F(x,y) \in \mathbb{Z}[x,y]$. Recall that a binary form is a polynomial in two variables which is homogeneous, and $h$ is a non-zero integer. Typically, since $F$ is homogeneous, it is convenient to allow $h$ to be positive and further require that the inputs $(x,y)$ be relatively prime integers.

The fact that binary forms always split into linear forms over $\overline{\mathbb{Q}}$ is the reason why substantially better results can be proved for binary forms rather than general binary polynomials, and homogeneous polynomials in a greater number of variables.

Nevertheless, has any progress been made for equations of the form $$\displaystyle F(x_1, \cdots, x_n) = h,$$ where $F(x_1, \cdots, x_n) \in \mathbb{Z}[x_1, \cdots, x_n]$ is a homogeneous polynomial in $n$ variables?

I know that for $n = 3$ a 2009 paper by D.R. Heath-Brown has made progress on counting solutions inside the box $|x_i| \leq B$, with $h$ relatively small compared to $B$. O. Marmon generalized Heath-Brown's argument to the $n = 4$ case. Both employ a version of the determinant method pioneered by Bombieri and Pila in their 1989 paper. Both results have the deficit that the range of $h$ is fairly restricted. In particular, they are not able to prove the analogue of the following result for binary forms: if we wish to count the number of pairs $(x,y)$ such that $1 \leq F(x,y) \leq X$ with the assumption that $F(1,0) > 0$, then the exact order of magnitude is $X^{2/d}$ where $d = \deg(F)$.

Are there any other attempts to deal with the number of solutions to the higher dimensional analogue of Thue's equation? Any references would be appreciated.

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    $\begingroup$ Typically, lower bounds for linear forms in logarithms will not be of help here, since, for $n \geq 3$, such forms are usually irreducible. If the form decomposes, one can say something -- there are many papers of Gyory on the subject. $\endgroup$ – Mike Bennett Feb 28 '14 at 7:51
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    $\begingroup$ If the hypersurface $F(x_1,\ldots,x_n)=hx_0^d$ is non-singular, or more generally has only transversal intersections, then Vojta's conjecture implies that the solutions to $F=h$ with $\mathbf{x}\in\mathbb{Z}^n$ lie on a proper subvariety if $d\ge n+1$, and similarly for the solutions $\mathbf{x}\in\mathbb{Q}^n$ if $d\ge n+2$. (For $n=2$, these are, of course, due to Thue and Faltings, respectively.) $\endgroup$ – Joe Silverman Feb 28 '14 at 14:35
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Here is an example which illustrates that in general things are incredibly hard. Consider the equation:

$$x_1^3 + x_2^3 + x_3^3 = h.$$

Here are two famous open problems:

  1. For which $h$ does this equation have a solution in the integers?
  2. Is there any $h$, which is not a cube, for which the equation has infinitely many solutions in the integers?

With regards to 1., simple congruence conditions show that there is never a solution if $h \equiv 4,5 \mod 9$. It is conjectured that the converse holds. I believe the simplest case where this is open is the case $h=33$ (or possibly $h=30$?). Note that many computer experiments have been performed on various $h$ looking for solutions and when solutions are found, they often have incredibly large height. Just search google for "sums of three cubes".

Also for 2. one can show that there are infinitely many solutions if $d$ is a cube.

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  • $\begingroup$ I would suspect that in the higher dimensional case the number of solutions should be infinite more often than the two dimensional case (where $\deg F \geq 3$ is sufficient for the number of solutions to be at most finite), but there should still be some way to give an upper bound in terms of the height of the solutions or something. $\endgroup$ – Stanley Yao Xiao Feb 28 '14 at 13:41
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    $\begingroup$ The case $h=30$ is solved: $30 = 2220422932^3 − 283059965^3 − 2218888517^3$ in Elkies paper on page $18$ here: arxiv.org/pdf/math/0005139v1.pdf. $\endgroup$ – Dietrich Burde Feb 28 '14 at 16:23

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