View source for Linear form in logarithms


''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 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]].

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 $|L| > \exp(-\espilon B)$. 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]]]).

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|Algebraic number]]). 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 [[#References|[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|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====
<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====



====References====
<table>
<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">[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>