User:Chenxlee/Baker

In transcendence theory, Baker's theorem is a far-reaching result concerning the linear independence of logarithms of algebraic numbers. The result, proved by Alan Baker in the 1960s, subsumed many earlier results in transcendental number theory and solved a problem posed by Alexander Gelfond nearly fifteen years earlier.

History

To simplify notation we introduce the set L of logarithms of nonzero algebraic numbers, that is

$L=\{\lambda \in \mathbb {C} \,:\,e^{\lambda }\in {\overline {\mathbb {Q} }}^{\times }\}.$

Using this notation several results in transcendental number theory become much easier to state, for example the Hermite–Lindemann theorem becomes the statement that any nonzero element of L is transcendental.

In 1934, Alexander Gelfond and Theodor Schneider independently proved the Gelfond–Schneider theorem. This result is usually stated as: if a is algebraic and not equal to 0 or 1, and if b is algebraic and irrational, then a^b is transcendental. Equivalently, though, it says that if λ₁ and λ₂ are elements of L that are linearly independent over the rational numbers, then they are linearly independent over the algebraic numbers. So if λ₁ and λ₂ are elements of L and λ₂ isn't zero, then the quotient λ₁/λ₂ is either a rational number or transcendental, it can't be an algebraic irrational number like √2.

Although proving this result of "rational linear independence implies algebraic linear independence" for two elements of L was sufficient for his and Schneider's result, Gelfond felt that it was crucial to extend this result to arbitrarily many elements of L. Indeed, from page 177 of the translation of his book:

…one may assume … that the most pressing problem in the theory of transcendental numbers is the investigation of the measures of transcendence of finite sets of logarithms of algebraic numbers.

This problem was solved fourteen years later by Alan Baker and has since had numerous applications not only to transcendence theory but in algebraic number theory and the study of Diophantine equations as well. Baker received the Fields medal in 1970 for both this work and his applications of it to Diophantine equations.

Statement

With the above notation, Baker's theorem is a nonhomogeneous generalisation of the Gelfond–Schneider theorem. Specifically it states:

If λ₁,…,λ_n are elements of L that are linearly independent over the rational numbers, then 1, λ₁,…,λ_n are linearly independent over the algebraic numbers.

Just as the Gelfond–Schneider theorem is equivalent to the statement about the transcendence of numbers of the form a^b, so too is Baker's theorem equivalent to the transcendence of numbers of the form

$a_{1}^{b_{1}}\cdots a_{n}^{b_{n}},$

where the b_i are all algebraic, irrational, and 1, b₁,…,b_n are linearly independent over the rationals, and the a_i are all algebraic and not 0 or 1.

Corollaries

As mentioned above, the theorem includes numerous earlier transcendence results concerning the exponential function, such as the Hermite–Lindemann theorem and Gelfond–Schneider theorem. It is not quite as encompassing as the still unproven Schanuel's conjecture, and does not imply the six exponentials theorem nor, clearly, the still open four exponentials conjecture.

The main reason Gelfond desired an extension of his result was not just for a slew of new transcendental numbers. In 1935 he used the tools he had developed to prove the Gelfond-Schneider theorem to derive a lower bound for the quantity

$|\beta _{1}\lambda _{1}+\beta _{2}\lambda _{2}|\,$

where β₁ and β₂ are algebraic and λ₁ and λ₂ are in L. Baker's proof gave lower bounds for quantities like the above but with arbitrarily many terms, and he could use these bounds to develop effective means of tackling Diophantine equations and to solve Gauss' class number problem.

Extensions

Baker's theorem grants us the linear independence over the algebraic numbers of logarithms of algebraic numbers. This is weaker than proving their algebraic independence. So far no progress has been made on this problem at all. It has been conjectured that if λ₁,…,λ_n are elements of L that are linearly independent over the rational numbers, then they are algebraically independent too. This is a special case of Schanuel's conjecture, but so far it remains to be proved that there even exist two algebraic numbers whose logarithms are algebraically independent.

References

• Baker, Alan (1975). Transcendental number theory. Cambridge University Press. ISBN 0-521-20461-5.
• Gelfond, Alexander (1952). Transcendental and algebraic numbers. Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow.
• Sprindžuk, Vladimir Gennadievich (1993) [Russian original published in 1982]. Classical Diophantine equations. Lecture notes in mathematics. Translated by Ross Talent and Alf van der Poorten. Berlin: Springer.
• Waldschmidt, Michel (2000). Diophantine approximation on linear algebraic groups. Springer. ISBN 3-540-66785-7.