# Difference between revisions of "Thue method"

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− | A method in the theory of [[Diophantine approximations|Diophantine approximations]], created by A. Thue [[#References|[1]]] in connection with the problem of approximating algebraic numbers (cf. [[Algebraic number|Algebraic number]]) by rational numbers: Find a quantity | + | {{TEX|done}} |

+ | A method in the theory of [[Diophantine approximations|Diophantine approximations]], created by A. Thue [[#References|[1]]] in connection with the problem of approximating algebraic numbers (cf. [[Algebraic number|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 | + | 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 | + | 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 | + | 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 | + | The best possible estimate of the size of $\nu$ in \ref{1} was obtained by K.F. Roth [[#References|[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 geometry]]; [[Diophantine set|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|Diophantine approximation, problems of effective]]. | See also [[Diophantine approximation, problems of effective|Diophantine approximation, problems of effective]]. | ||

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====Comments==== | ====Comments==== | ||

− | Thue's method has been extended by C.L. Siegel to obtain < | + | Thue's method has been extended by C.L. Siegel to obtain $\nu<2\sqrt n$. For a good exposition, see [[#References|[a1]]]. This method, known as the Thue–Siegel method, has recently met with spectacular success when P. Vojta [[#References|[a2]]] showed how it could be used to give a new proof of the [[Mordell conjecture|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|Thue–Siegel–Roth theorem]]. |

====References==== | ====References==== | ||

<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> E. Landau, "Vorlesungen über Zahlentheorie" , Chelsea, reprint (1969)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> P. Vojta, "Siegel's theorem in the compact case" ''Ann. of Math.'' (Forthcoming)</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> E. Landau, "Vorlesungen über Zahlentheorie" , Chelsea, reprint (1969)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> P. Vojta, "Siegel's theorem in the compact case" ''Ann. of Math.'' (Forthcoming)</TD></TR></table> |

## Revision as of 15:02, 2 August 2014

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) |

**How to Cite This Entry:**

Thue method.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Thue_method&oldid=13235