Clemens Heuberger - Thue equations

Diophantine 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.

Thue equations

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: (dvi, dvi.gz, ps, ps.gz, pdf, pdf.gz)

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