# Thue method

2010 Mathematics Subject Classification: Primary: 11J68 [MSN][ZBL]

A method in the theory of Diophantine approximations, created by A. Thue  in connection with the problem of approximating algebraic numbers (cf. Algebraic number) by rational numbers: Find a quantity $\nu=\nu(n)$ such that for each algebraic number $\alpha$ of degree $n$ the inequality

$$\left|\alpha-\frac pq\right|<\frac{1}{q^{\nu+\epsilon}}\label{1}\tag{1}$$

has a finite number of solutions in rational integers $p$ and $q$, $q>0$, for any $\epsilon>0$, and an infinite number of solutions for any $\epsilon<0$.

Thue proved that $\nu\leq(n/2)+1$. Thue's method is based on properties of a special polynomial $f(x,y)$ of two variables $x,y$ with integer coefficients, and the hypothesis that there exist two solutions of \eqref{1} for $\nu\leq(n/2)+1$ with sufficiently large values of $q$. Thue's theorem has many important applications in number theory. In particular, it implies that the Diophantine equation

$$F(x,y)=m,\label{2}\tag{2}$$

where $F(x,y)$ is an irreducible form in the variables $x$ and $y$ with integer coefficients and of degree $n\leq3$, while $m$ is an integer, cannot have more than a finite number of solutions in integers $x$ and $y$.

The best possible estimate of the size of $\nu$ in \eqref{1} was obtained by K.F. Roth , by generalizing Thue's method to the case of a polynomial in any number of variables, similar to the polynomial $f(x,y)$, and making use of the large number of solutions of \eqref{1}. The result, called the Thue–Siegel–Roth theorem, states that $\nu=2$ for any $n\geq2$. Thue's method has a generalization to the case of approximation of algebraic numbers by algebraic numbers. Thue's method is a general method for proving the finiteness of the number of integer points on a wide class of curves on algebraic varieties (see Diophantine geometry; Diophantine set). Apart from this, Thue's method has essential deficiencies: it is a non-effective method in the sense that it does not provide an answer to the question whether there exist in fact solutions of the inequalities \eqref{1} or the corresponding equations \eqref{2} that can be made use of in the proofs. Thus, Thue's method, in solving the question on the finiteness of the number of solutions of equation \eqref{2}, does not provide the possibility of determining whether an actual equation of this type is solvable and what the sizes of the estimates of the solutions $x,y$ are in their dependence on $F$.

How to Cite This Entry:
Thue method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Thue_method&oldid=44688
This article was adapted from an original article by A.F. Lavrik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article