Thue method

A method in the theory of Diophantine approximations, created by A. Thue [1] 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

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

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 \ref{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

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

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 \ref{1} was obtained by K.F. Roth [2], 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 \ref{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 \ref{1} or the corresponding equations \ref{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 \ref{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 $F$.

See also Diophantine approximation, problems of effective.

References

[1]	A. Thue, "Ueber Annäherungswerte algebraischer Zahlen" J. Reine Angew. Math. , 135 (1909) pp. 284–305
[2]	K.F. Roth, "Rational approximation to algebraic numbers" Mathematika , 2 : 1 (1955) pp. 1–20
[3]	, Problems in the theory of Diophantine approximations , Moscow (1974) (In Russian; translated from English)

Comments

Thue's method has been extended by C.L. Siegel to obtain $\nu<2\sqrt n$. For a good exposition, see [a1]. This method, known as the Thue–Siegel method, has recently met with spectacular success when P. Vojta [a2] showed how it could be used to give a new proof of the Mordell conjecture. A considerable and very accessible simplification of this proof has been given by E. Bombieri. For other generalizations see Thue–Siegel–Roth theorem.

References

[a1]	E. Landau, "Vorlesungen über Zahlentheorie" , Chelsea, reprint (1969)
[a2]	P. Vojta, "Siegel's theorem in the compact case" Ann. of Math. (Forthcoming)