In transcendental number theory, a mathematical discipline, Baker's theorem gives a lower bound for the absolute value of linear combinations of logarithms of algebraic numbers.
Nearly fifteen years earlier, Alexander Gelfond had considered the problem with only integer coefficients to be of "extraordinarily great significance".
Using this notation, several results in transcendental number theory become much easier to state.
In 1934, Alexander Gelfond and Theodor Schneider independently proved the Gelfond–Schneider theorem.
The exponential function is multi-valued for complex exponents, and this applies to all of its values, which in most cases constitute infinitely many numbers.
Indeed, from Gel'fond (1960, p. 177): ...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.
Just as the Gelfond–Schneider theorem is equivalent to the statement about the transcendence of numbers of the form ab, so too Baker's theorem implies the transcendence of numbers of the form where the bi are all algebraic, irrational, and 1, b1, ..., bn are linearly independent over the rationals, and the ai are all algebraic and not 0 or 1.
are rational integers, the rightmost term log Ω can be deleted.
An explicit result by Baker and Wüstholz for a linear form Λ with integer coefficients yields a lower bound of the form where and d is the degree of the number field generated by the
Baker's proof of his theorem is an extension of the argument given by Gel'fond (1960, chapter III, section 4).
The main ideas of the proof are illustrated by the proof of the following qualitative version of the theorem of Baker (1966) described by Serre (1971): The precise quantitative version of Baker's theory can be proved by replacing the conditions that things are zero by conditions that things are sufficiently small throughout the proof.
The main idea of Baker's proof is to construct an auxiliary function
then repeatedly show that it vanishes to lower order at even more points of this form.
Finally the fact that it vanishes (to order 1) at enough points of this form implies using Vandermonde determinants that there is a multiplicative relation between the numbers ai.
The constants L, h, and M have to be carefully adjusted so that the next part of the proof works, and are subject to some constraints, which are roughly: The constraints can be satisfied by taking h to be sufficiently large, M to be some fixed power of h, and L to be a slightly smaller power of h. Baker took M to be about h2 and L to be about h2−1/2n.
The linear relation between the logarithms of the α's is used to reduce L slightly; roughly speaking, without it the condition Ln must be larger than about Mn−1h would become Ln must be larger than about Mnh, which is incompatible with the condition that L is somewhat smaller than M. The next step is to show that Φ vanishes to slightly smaller order at many more points of the form
for integers l. This idea was Baker's key innovation: previous work on this problem involved trying to increase the number of derivatives that vanish while keeping the number of points fixed, which does not seem to work in the multivariable case.
Then one shows that derivatives of Φ at this point are given by algebraic integers times known constants.
Combining these two ideas implies that Φ vanishes to slightly smaller order at many more points
This part of the argument requires that Φ does not increase too rapidly; the growth of Φ depends on the size of L, so requires a bound on the size of L, which turns out to be roughly that L must be somewhat smaller than M. More precisely, Baker showed that since Φ vanishes to order M at h consecutive integers, it also vanishes to order M/2 at h1+1/8n consecutive integers 1, 2, 3, ....
Repeating this argument J times shows that Φ vanishes to order M/2J at h1+J/8n points, provided that h is sufficiently large and L is somewhat smaller than M/2J.
can be written as: Therefore as l varies we have a system of (L + 1)n homogeneous linear equations in the (L + 1)n unknowns which by assumption has a non-zero solution, which in turn implies the determinant of the matrix of coefficients must vanish.
Baker (1966) in fact gave a quantitative version of the theorem, giving effective lower bounds for the linear form in logarithms.
But these equations for the first (L+1)n derivatives again give a homogeneous set of linear equations for the coefficients p, so the determinant is zero, and is again a Vandermonde determinant, this time for the numbers λ1 log α1 + ⋯ + λn log αn.
Baker (1967b) gave an inhomogeneous version of the theorem, showing that is nonzero for nonzero algebraic numbers β0, ..., βn, α1, ..., αn, and moreover giving an effective lower bound for it.
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 where β1 and β2 are algebraic and λ1 and λ2 are in
[2] 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.
Indeed, Baker's theorem rules out linear relations between logarithms of algebraic numbers unless there are trivial reasons for them; the next most simple case, that of ruling out homogeneous quadratic relations, is the still open four exponentials conjecture.