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".[1] The result, proved by, subsumed many earlier results in transcendental number theory. Baker used this to prove the transcendence of many numbers, to derive effective bounds for the solutions of some Diophantine equations, and to solve the class number problem of finding all imaginary quadratic fields with class number 1.
To simplify notation, let
L
\Complex
\overline{\Q}
\Q
L
In 1934, Alexander Gelfond and Theodor Schneider independently proved the Gelfond–Schneider theorem. This result is usually stated as: if
a
b
ab
λ1,λ2\inL
λ1,λ2\inL
λ2
λ1/λ2
\sqrt2
Although proving this result of "rational linear independence implies algebraic linear independence" for two elements of
L
L.
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.
With the above notation, Baker's theorem is a nonhomogeneous generalization of the Gelfond–Schneider theorem. Specifically it states:
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
b1 | |
a | |
1 |
…
bn | |
a | |
n |
,
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.
also gave several versions with explicit constants. For example, if
\exp(λj)=\alphaj
Aj\ge4
\betaj
B\ge4
Λ=\beta0+\beta1λ1+ … +\betanλn
is either 0 or satisfies
log|Λ|>(16nd)200n\Omega\left(log\Omega-loglogAn\right)(logB+log\Omega)
where
\Omega=logA1logA2 … logAn
and the field generated by
\alphai
\betai
\betaj
An explicit result by Baker and Wüstholz for a linear form Λ with integer coefficients yields a lower bound of the form
log|Λ|>-Ch(\alpha1)h(\alpha2) … h(\alphan)log\left(max\left\{|\beta1|,\ldots,|\betan|\right\}\right),
where
C=18(n+1)!nn+1(32d)n+2log(2nd),
and d is the degree of the number field generated by the
\alphai.
Baker's proof of his theorem is an extension of the argument given by . The main ideas of the proof are illustrated by the proof of the following qualitative version of the theorem of described by :
If the numbers
2\pii,loga1,\ldots,logan
a1,\ldots,an,
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.
\Phi(z1,\ldots,zn-1)
z1= … =zn-1=l,
Assume there is a relation
\beta1log\alpha1+ … +\betan-1log\alphan-1=log\alphan
for algebraic numbers α1, ..., αn, β1, ..., βn−1. The function Φ is of the form
\Phi(z1,\ldots,zn-1)=
L … | |
\sum | |
λ1=0 |
L | |
\sum | |
λn=0 |
p(λ1,\ldots,λn)
(λ1+λn\beta1)z1 | |
\alpha | |
1 |
…
(λn-1+λn\betan-1)zn-1 | |
\alpha | |
n-1 |
The integer coefficients p are chosen so that they are not all zero and Φ and its derivatives of order at most some constant M vanish at
z1= … =zn-1=l,
l
0\leql\leqh
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
z1= … =zn-1=l
z1= … =zn-1=l.
One then takes J large enough that:
| ||||||
h |
>(L+1)n.
(J larger than about 16n will do if h2 > L) so that:
\foralll\in\left\{1,2,\ldots,(L+1)n\right\}: \Phi(l,\ldots,l)=0.
By definition
\Phi(l,\ldots,l)=0
L | |
\sum | |
λ1=0 |
…
L | |
\sum | |
λn=0 |
p(λ1,\ldots,λn)
λ1l | |
\alpha | |
1 |
…
λnl | |
\alpha | |
n |
=0.
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. However this matrix is a Vandermonde matrix and the formula for the determinant of such a matrix forces an equality between two of the values:
λ1 | |
\alpha | |
1 |
…
λn | |
\alpha | |
n |
so
\alpha1,\ldots,\alphan
2\pii,log\alpha1,\ldots,log\alphan
in fact gave a quantitative version of the theorem, giving effective lower bounds for the linear form in logarithms. This is done by a similar argument, except statements about something being zero are replaced by statements giving a small upper bound for it, and so on.
showed how to eliminate the assumption about 2πi in the theorem. This requires a modification of the final step of the proof. One shows that many derivatives of the function
\phi(z)=\Phi(z,\ldots,z)
gave an inhomogeneous version of the theorem, showing that
\beta0+\beta1log\alpha1+ … +\betanlog\alphan
is nonzero for nonzero algebraic numbers β0, ..., βn, α1, ..., αn, and moreover giving an effective lower bound for it. The proof is similar to the homogeneous case: one can assume that
\beta0+\beta1log\alpha1+ … +\betan-1log\alphan-1=log\alphan
and one inserts an extra variable z0 into Φ as follows:
\Phi(z0,\ldots,zn-1)=
L | |
\sum | |
λ0=0 |
…
L | |
\sum | |
λn=0 |
p(λ0,\ldots,λn)
λ0 | |
z | |
0 |
λn\beta0z0 | |
e |
(λ1+λn\beta1)z1 | |
\alpha | |
1 |
(λn-1+λn\betan-1)zn-1 | |
… \alpha | |
n-1 |
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
|\beta1λ1+\beta2λ2|
where β1 and β2 are algebraic and λ1 and λ2 are in
L
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, ..., λn are elements of
L
Similarly, extending the result to algebraic independence but in the p-adic setting, and using the p-adic logarithm function, remains an open problem. It is known that proving algebraic independence of linearly independent p-adic logarithms of algebraic p-adic numbers would prove Leopoldt's conjecture on the p-adic ranks of units of a number field.