Skolem’s Method to solve Thue-Siegel Equations

We want to prove that equation has finitely many solution in integers for any homogeneous polynomial of degree at least

One standard way to prove the result to use diophantine approximation and polynomial method. We can create very good rational approximations to the roots of the polynomial if there are solutions to the equation, and then by polynomial method argue bounds on the number of solutions. (One uses auxillary polynomials that forces too good of approximations)

But the diophantine approximation method’s are ineffective. We have no control of the size of the rational approximations and the heights of the solutions.

Skolem’s p-adic method write the equation as solving a norm equation in a number field. Then the finiteness of ideals of bounded norm reduces the problem to solving equations with units. We have to prove that as the units vary, only finitely many choices give integer solutions. Localizing modulo a prime, and taking logarithms the equation can be seen over p-adic integers. Expanding the power series of logarithms p-adically, we get equations that can be shown to have only finitely many p-adic integral solutions by Hensel type argument.

http://users.math.uoc.gr/~tzanakis/Papers/p-adicMethodSkolem.pdf

There is also Bakers method on linear forms in logarithms. Skolem’s method and Baker’s method are both effective.