## Generalizations of Thue’s equation

It seems that there is significant interest vested in questions regarding the Thue equation, namely the equation $$F(x,y) = h,$$ where $F(x,y) = a_r x^r + a_{r-1}x^{r-1}y + \cdots + a_0 y^r$ for $a_0, a_1, \cdots, a_r \in \mathbb{Z}$ and $x,y$ integers with $h$ a fixed integer. One technique used to study this problem is to consider actions of $\text{GL}(2, \mathbb{Z})$ or $\text{SL}(2, \mathbb{Z})$ on $F$, which changes the diophantine approximation properties of $F$ but preserves the number of solutions (see E. Bombieri and W.M. Schmidt, "On Thue's Equation", Invent. Math, 88., (1987), 69-81.)

My question is has anyone attempted to use actions of $\text{GL}(n, \mathbb{Z})$ or $\text{SL}(n, \mathbb{Z})$ to study equations of the type $F(x_1, \cdots, x_n) = h$?

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 Certainly when the degree is 2. – Charles Matthews Jun 10 2011 at 19:42