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''of algebraic numbers''
 
''of algebraic numbers''
  
 
An expression of the form
 
An expression of the form
 +
$$
 +
L = \beta_1 \log \alpha_1 + \cdots + \beta_n \log \alpha_n \ .
 +
$$
  
<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/l/l059/l059220/l0592201.png" /></td> </tr></table>
+
Effective lower bounds for $|L|$, under the assumption that the numbers $\alpha_1,\ldots,\alpha_n$, $\beta_1,\ldots,\beta_n$  are [[Rational number|rational]] or [[algebraic number]]s and $\log\alpha_1,\ldots,\log\alpha_n$, with fixed branches of the logarithms, are linearly independent over the field $\mathbb{Q}$, play an important role in [[number theory]], with applications to [[Diophantine equations]].
  
Effective lower bounds for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059220/l0592202.png" />, under the assumption that the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059220/l0592203.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059220/l0592204.png" /> are rational or algebraic numbers and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059220/l0592205.png" />, with fixed branches of the logarithms, are linearly independent over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059220/l0592206.png" />, play an important role in number theory.
+
When $\beta_1,\ldots,\beta_n$ are rational, say $\beta_I = p_i/q_i$, the inequality $|L| > \exp(- c_1 B)$ holds, where $B = \max \{|p_i|,|q_i|\}$ and $c_1>0$ depends only on the numbers $\alpha_1,\ldots,\alpha_n$. The methods by means of which non-trivial lower bounds for $|L|$ are established belong to the theory of transcendental numbers. In the case $n=2$ a number of inequalities, true for $B$, better than an effectively computable bound, were obtained by A.O. Gel'fond in 1935–1949. The best of them has the form $|L| > \exp(-\log^{2+\epsilon}B)$.
  
When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059220/l0592207.png" /> are rational, the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059220/l0592208.png" /> holds, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059220/l0592209.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059220/l05922010.png" /> depends only on the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059220/l05922011.png" />. The methods by means of which non-trivial lower bounds for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059220/l05922012.png" /> are established belong to the theory of transcendental numbers. In the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059220/l05922013.png" /> a number of inequalities, true for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059220/l05922014.png" />, better than an effectively computable bound, were obtained by A.O. Gel'fond in 1935–1949. The best of them has the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059220/l05922015.png" />.
+
In 1948 he proved that for any $n$ and for all sufficiently large $B$ one has $|L| > \exp(-\epsilon B)$. The latter result was, however, only an existence theorem, and a bound for $B$, beyond which this inequality was satisfied, could not be determined from the proof. Effective bounds for $|L|$ for any $n$ were obtained in 1966 by A. Baker (see [[#References|[2]]]).
 
 
In 1948 he proved that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059220/l05922016.png" /> and for all sufficiently large <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059220/l05922017.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059220/l05922018.png" />. The latter result was, however, only an existence theorem, and a bound for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059220/l05922019.png" />, starting with which this inequality was satisfied, could not be determined from the proof. Effective bounds for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059220/l05922020.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059220/l05922021.png" /> were obtained in 1966 by A. Baker (see [[#References|[2]]]) on the basis of Gel'fond's method.
 
 
 
Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059220/l05922022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059220/l05922023.png" /> are non-zero algebraic numbers with height and degree not exceeding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059220/l05922024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059220/l05922025.png" />, respectively, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059220/l05922026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059220/l05922027.png" /> (cf. [[Algebraic number|Algebraic number]]). Suppose also that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059220/l05922028.png" /> and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059220/l05922029.png" /> are the principal values of the logarithms. If there are rational integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059220/l05922030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059220/l05922031.png" />, such that
 
 
 
<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/l/l059/l059220/l05922032.png" /></td> </tr></table>
 
  
 +
Suppose that $n \ge 2$ and $\alpha_1,\ldots,\alpha_n$ are non-zero algebraic numbers with height and degree not exceeding $A$ and $d$, respectively, where $A \ge 4$, $d \ge 4$ (cf. [[Algebraic number]] and [[Height, in Diophantine geometry]]). Suppose also that $0 < \epsilon < 1$ and that the $\log \alpha_i$ are the principal values of the logarithms. If there are rational integers $b_1,\ldots,b_n$, $|b_i| < B$, such that
 +
$$
 +
0 < | b_1 \log \alpha_1 + \cdots + b_n \log \alpha_n | < \exp(-\epsilon B)
 +
$$
 
then
 
then
 +
$$
 +
B < (4^{n^2} \epsilon^{-1} d^{2n} \log A)^{(2n+1)^2} \ .
 +
$$
  
<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/l/l059/l059220/l05922033.png" /></td> </tr></table>
 
  
In connection with various problems a large number of effective bounds for linear forms in logarithms have been obtained. A bound for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059220/l05922034.png" /> in terms of powers of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059220/l05922035.png" /> was first obtained in 1968 by N.I. Fel'dman [[#References|[3]]].
+
In connection with various problems a large number of effective bounds for linear forms in logarithms have been obtained. A bound for$|L|$ in terms of powers of $B$ was first obtained in 1968 by N.I. Fel'dman [[#References|[3]]].
 
 
Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059220/l05922036.png" />, that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059220/l05922037.png" /> are algebraic numbers and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059220/l05922038.png" />, with fixed branches of the logarithms, are linearly independent over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059220/l05922039.png" />. There are effective constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059220/l05922040.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059220/l05922041.png" /> such that for any algebraic numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059220/l05922042.png" /> with height not exceeding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059220/l05922043.png" /> the inequality
 
 
 
<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/l/l059/l059220/l05922044.png" /></td> </tr></table>
 
 
 
holds (the constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059220/l05922045.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059220/l05922046.png" /> can be given explicitly in terms of the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059220/l05922047.png" /> and powers of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059220/l05922048.png" />).
 
 
 
By means of bounds for linear forms in logarithms of algebraic numbers, bounds have been obtained for solutions of various classes of [[Diophantine equations|Diophantine equations]] (Thue equations, hyper-elliptic equations, equations given by curves of genus 1, etc.). Estimates of linear forms in logarithms have made it possible to determine bounds for the discriminants of imaginary quadratic fields with class numbers 1 and 2. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059220/l05922049.png" />-adic analogues of theorems giving bounds for linear forms in logarithms of algebraic numbers are also used in number theory.
 
 
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.O. Gel'fond,  "Transcendental and algebraic numbers" , Dover, reprint  (1960)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Baker,  "Effective methods in the theory of numbers" , ''Proc. Internat. Congress Mathematicians (Nice, 1970)'' , '''1''' , Gauthier-Villars  (1971)  pp. 19–26</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N.I. Fel'dman,  "An improvement of the estimate for a linear form in the logarithms of algebraic numbers"  ''Math. USSR Sb.'' , '''6'''  (1968)  pp. 393–406  ''Mat. Sb.'' , '''77''' :  3  (1968)  pp. 423–436</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N.I. Fel'dman,  "An effective refinement of the exponent in Liouville's theorem"  ''Math. USSR Izv.'' , '''5''' :  5  (1971)  pp. 985–1002  ''Izv. Akad. Nauk. SSSR Ser. Mat.'' , '''35''' :  5  (1971)  pp. 973–990</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> , ''Current problems of analytic number theory'' , Minsk  (1974)  (In Russian)</TD></TR></table>
 
 
 
 
 
 
 
====Comments====
 
  
 +
Suppose that $n \ge 2$, that $\alpha_1,\ldots,\alpha_n$ are algebraic numbers and that $\log\alpha_1,\ldots,\alpha_n$, with fixed branches of the logarithms, are linearly independent over $\mathbb Q$. There are effective constants $c_2 > 0$, $\kappa_1 > 0$, such that for any algebraic numbers $\beta_0, \beta_1,\ldots,\beta_n$ with height not exceeding $|B|$ the inequality
 +
$$
 +
| \beta_0 + \beta_1 \log \alpha_1 + \cdots + \beta_n \log \alpha_n | > c_2 B^{\kappa_1}
 +
$$
 +
holds (the constants $c_2$ and $\kappa_1$ can be given explicitly in terms of the numbers $\alpha_1,\ldots,\alpha_n$ and powers of $\beta_0, \beta_1,\ldots,\beta_n$).
  
 +
By means of bounds for linear forms in logarithms of algebraic numbers, bounds have been obtained for solutions of various classes of [[Diophantine equations|Diophantine equations]] (Thue equations, hyper-elliptic equations, equations given by curves of genus 1, etc.). Estimates of linear forms in logarithms have made it possible to determine bounds for the [[discriminant]]s of imaginary quadratic fields with class numbers 1 and 2. $p$-adic analogues of theorems giving bounds for linear forms in logarithms of algebraic numbers are also used in number theory.
  
 
====References====
 
====References====
 
<table>
 
<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top">  A.O. Gel'fond,  "Transcendental and algebraic numbers" , Dover, reprint  (1960)  (Translated from Russian)</TD></TR>
 +
<TR><TD valign="top">[2]</TD> <TD valign="top">  A. Baker,  "Effective methods in the theory of numbers" , ''Proc. Internat. Congress Mathematicians (Nice, 1970)'' , '''1''' , Gauthier-Villars  (1971)  pp. 19–26</TD></TR>
 +
<TR><TD valign="top">[3]</TD> <TD valign="top">  N.I. Fel'dman,  "An improvement of the estimate for a linear form in the logarithms of algebraic numbers"  ''Math. USSR Sb.'' , '''6'''  (1968)  pp. 393–406  ''Mat. Sb.'' , '''77''' :  3  (1968)  pp. 423–436</TD></TR>
 +
<TR><TD valign="top">[4]</TD> <TD valign="top">  N.I. Fel'dman,  "An effective refinement of the exponent in Liouville's theorem"  ''Math. USSR Izv.'' , '''5''' :  5  (1971)  pp. 985–1002  ''Izv. Akad. Nauk. SSSR Ser. Mat.'' , '''35''' :  5  (1971)  pp. 973–990</TD></TR>
 +
<TR><TD valign="top">[5]</TD> <TD valign="top"> , ''Current problems of analytic number theory'' , Minsk  (1974)  (In Russian)</TD></TR>
 
<TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Baker,  "Transcendental number theory" , Cambridge Univ. Press  (1975)</TD></TR>
 
<TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Baker,  "Transcendental number theory" , Cambridge Univ. Press  (1975)</TD></TR>
 
<TR><TD valign="top">[a2]</TD> <TD valign="top">  T.N. Shorey,  R. Tijdeman,  "Exponential Diophantine equations" , Cambridge Univ. Press  (1986)</TD></TR>
 
<TR><TD valign="top">[a2]</TD> <TD valign="top">  T.N. Shorey,  R. Tijdeman,  "Exponential Diophantine equations" , Cambridge Univ. Press  (1986)</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top">Alan Baker, Gisbert Wüstholz, "Logarithmic Forms and Diophantine Geometry", New Mathematical Monographs '''9''', Cambridge University Press (2007), ISBN 978-0-521-88268-2 </TD></TR>
+
<TR><TD valign="top">[a3]</TD> <TD valign="top">Alan Baker, Gisbert Wüstholz, "Logarithmic Forms and Diophantine Geometry", New Mathematical Monographs '''9''', Cambridge University Press (2007), {{ISBN|978-0-521-88268-2}} </TD></TR>
 
</table>
 
</table>
 +
 +
[[Category:Number theory]]

Latest revision as of 08:31, 23 November 2023

of algebraic numbers

An expression of the form $$ L = \beta_1 \log \alpha_1 + \cdots + \beta_n \log \alpha_n \ . $$

Effective lower bounds for $|L|$, under the assumption that the numbers $\alpha_1,\ldots,\alpha_n$, $\beta_1,\ldots,\beta_n$ are rational or algebraic numbers and $\log\alpha_1,\ldots,\log\alpha_n$, with fixed branches of the logarithms, are linearly independent over the field $\mathbb{Q}$, play an important role in number theory, with applications to Diophantine equations.

When $\beta_1,\ldots,\beta_n$ are rational, say $\beta_I = p_i/q_i$, the inequality $|L| > \exp(- c_1 B)$ holds, where $B = \max \{|p_i|,|q_i|\}$ and $c_1>0$ depends only on the numbers $\alpha_1,\ldots,\alpha_n$. The methods by means of which non-trivial lower bounds for $|L|$ are established belong to the theory of transcendental numbers. In the case $n=2$ a number of inequalities, true for $B$, better than an effectively computable bound, were obtained by A.O. Gel'fond in 1935–1949. The best of them has the form $|L| > \exp(-\log^{2+\epsilon}B)$.

In 1948 he proved that for any $n$ and for all sufficiently large $B$ one has $|L| > \exp(-\epsilon B)$. The latter result was, however, only an existence theorem, and a bound for $B$, beyond which this inequality was satisfied, could not be determined from the proof. Effective bounds for $|L|$ for any $n$ were obtained in 1966 by A. Baker (see [2]).

Suppose that $n \ge 2$ and $\alpha_1,\ldots,\alpha_n$ are non-zero algebraic numbers with height and degree not exceeding $A$ and $d$, respectively, where $A \ge 4$, $d \ge 4$ (cf. Algebraic number and Height, in Diophantine geometry). Suppose also that $0 < \epsilon < 1$ and that the $\log \alpha_i$ are the principal values of the logarithms. If there are rational integers $b_1,\ldots,b_n$, $|b_i| < B$, such that $$ 0 < | b_1 \log \alpha_1 + \cdots + b_n \log \alpha_n | < \exp(-\epsilon B) $$ then $$ B < (4^{n^2} \epsilon^{-1} d^{2n} \log A)^{(2n+1)^2} \ . $$


In connection with various problems a large number of effective bounds for linear forms in logarithms have been obtained. A bound for$|L|$ in terms of powers of $B$ was first obtained in 1968 by N.I. Fel'dman [3].

Suppose that $n \ge 2$, that $\alpha_1,\ldots,\alpha_n$ are algebraic numbers and that $\log\alpha_1,\ldots,\alpha_n$, with fixed branches of the logarithms, are linearly independent over $\mathbb Q$. There are effective constants $c_2 > 0$, $\kappa_1 > 0$, such that for any algebraic numbers $\beta_0, \beta_1,\ldots,\beta_n$ with height not exceeding $|B|$ the inequality $$ | \beta_0 + \beta_1 \log \alpha_1 + \cdots + \beta_n \log \alpha_n | > c_2 B^{\kappa_1} $$ holds (the constants $c_2$ and $\kappa_1$ can be given explicitly in terms of the numbers $\alpha_1,\ldots,\alpha_n$ and powers of $\beta_0, \beta_1,\ldots,\beta_n$).

By means of bounds for linear forms in logarithms of algebraic numbers, bounds have been obtained for solutions of various classes of Diophantine equations (Thue equations, hyper-elliptic equations, equations given by curves of genus 1, etc.). Estimates of linear forms in logarithms have made it possible to determine bounds for the discriminants of imaginary quadratic fields with class numbers 1 and 2. $p$-adic analogues of theorems giving bounds for linear forms in logarithms of algebraic numbers are also used in number theory.

References

[1] A.O. Gel'fond, "Transcendental and algebraic numbers" , Dover, reprint (1960) (Translated from Russian)
[2] A. Baker, "Effective methods in the theory of numbers" , Proc. Internat. Congress Mathematicians (Nice, 1970) , 1 , Gauthier-Villars (1971) pp. 19–26
[3] N.I. Fel'dman, "An improvement of the estimate for a linear form in the logarithms of algebraic numbers" Math. USSR Sb. , 6 (1968) pp. 393–406 Mat. Sb. , 77 : 3 (1968) pp. 423–436
[4] N.I. Fel'dman, "An effective refinement of the exponent in Liouville's theorem" Math. USSR Izv. , 5 : 5 (1971) pp. 985–1002 Izv. Akad. Nauk. SSSR Ser. Mat. , 35 : 5 (1971) pp. 973–990
[5] , Current problems of analytic number theory , Minsk (1974) (In Russian)
[a1] A. Baker, "Transcendental number theory" , Cambridge Univ. Press (1975)
[a2] T.N. Shorey, R. Tijdeman, "Exponential Diophantine equations" , Cambridge Univ. Press (1986)
[a3] Alan Baker, Gisbert Wüstholz, "Logarithmic Forms and Diophantine Geometry", New Mathematical Monographs 9, Cambridge University Press (2007), ISBN 978-0-521-88268-2
How to Cite This Entry:
Linear form in logarithms. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Linear_form_in_logarithms&oldid=30203
This article was adapted from an original article by Yu.V. Nesterenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article