Clemens Heuberger - Thue equations
Since antiquity, many people try to solve equations over the
integers, Pythagoras for instance
described all integers being the sides of a rectangular triangle.
After Diophantus von
Alexandrien such equations are called diophantine equations.
Since that time, many mathematicians worked on this topic, such as
Fermat, Euler, Kummer, Siegel, and Wiles.
Among his 23 Problems, Hilbert
raised the question, whether there exists an algorithm to solve
any given polynomial diophantine equation; the negative answer
has been given by Matijasevic in 1970.
So the research interest in diophantine equations is to find
classes of such equations which can be solved.
In 1909, A. Thue
considered a special family of equations
F(X,Y) = m,
where F is an irreducible homogeneous form of degree n at least 3 and m is
a nonzero integer.
This type of equations is called after him since
then; he proved that such an equation only has a finite number of
solutions. His proof, however, is not constructive, so it does not lead to an
algorithm. Only with Baker's lower bounds for linear forms in logarithms of
algebraic numbers (1966--1968),
effective bounds for the solution of many diophantine equations can be given.
Since that time, many bounds have been improved and algorithms have been developed
to solve one single Thue-equation in reasonable time on a computer (see Bilu and Hanrot).
Parametrized Thue equations
In 1990, E. Thomas studied a parametrized family of cubic Thue equations:
It turns out that there exist only a few "trivial" solutions for
large values of the parameter.
Since that time, several concrete families and families of arbitrary
degree have been discussed. A survey can be found in the following forms:
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