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 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