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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092760/t0927601.png" /> such that for each algebraic number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092760/t0927602.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092760/t0927603.png" /> the inequality
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{{TEX|done}}{{MSC|11J68}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092760/t0927604.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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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 $\nu=\nu(n)$ such that for each algebraic number $\alpha$ of degree $n$ the inequality
  
has a finite number of solutions in rational integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092760/t0927605.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092760/t0927606.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092760/t0927607.png" />, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092760/t0927608.png" />, and an infinite number of solutions for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092760/t0927609.png" />.
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$$\left|\alpha-\frac pq\right|<\frac{1}{q^{\nu+\epsilon}}\label{1}\tag{1}$$
  
Thue proved that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092760/t09276010.png" />. Thue's method is based on properties of a special polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092760/t09276011.png" /> of two variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092760/t09276012.png" /> with integer coefficients, and the hypothesis that there exist two solutions of (1) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092760/t09276013.png" /> with sufficiently large values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092760/t09276014.png" />. Thue's theorem has many important applications in number theory. In particular, it implies that the Diophantine equation
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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 $\epsilon<0$.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092760/t09276015.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
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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
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092760/t09276016.png" /> is an irreducible form in the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092760/t09276017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092760/t09276018.png" /> with integer coefficients and of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092760/t09276019.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092760/t09276020.png" /> is an integer, cannot have more than a finite number of solutions in integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092760/t09276021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092760/t09276022.png" />.
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$$F(x,y)=m,\label{2}\tag{2}$$
  
The best possible estimate of the size of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092760/t09276023.png" /> in (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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092760/t09276024.png" />, and making use of the large number of solutions of (1). The result, called the Thue–Siegel–Roth theorem, states that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092760/t09276025.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092760/t09276026.png" />. 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 (1) or the corresponding equations (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 (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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092760/t09276027.png" /> are in their dependence on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092760/t09276028.png" />.
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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 $y$.
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The best possible estimate of the size of $\nu$ in \eqref{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 \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 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 \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 $F$.
  
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092760/t09276029.png" />. 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]].
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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>

Latest revision as of 15:17, 14 February 2020

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

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

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

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 [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 \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 $F$.

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
This article was adapted from an original article by A.F. Lavrik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article