Let F(x, y) be an irreducible binary form with integer coefficients and of degree at least 3. By a well known result of Thue, the equation F(x, y) = m has only finitely many solutions in integers x and y. I will discuss some old and new quantitative results on the number of solutions of such equations. I will also talk about my joint work with Manjul Bhargava, where we use these bounds to show that many equations of the shape F(x, y) = m have no solutions.
Links:
[1] https://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] https://www.mpim-bonn.mpg.de/node/3444
[3] https://www.mpim-bonn.mpg.de/node/246