Generalizations of Thue's equation - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T06:00:27Z http://mathoverflow.net/feeds/question/67465 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/67465/generalizations-of-thues-equation Generalizations of Thue's equation Stanley Yao Xiao 2011-06-10T18:54:54Z 2011-06-10T18:54:54Z <p>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.) </p> <p>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 $F(x_1, \cdots, x_n) = h$?</p>