History
To simplify notation, let be the set of logarithms to the base ''e'' of nonzeroStatement
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 : where the ''bi'' are all algebraic, irrational, and 1, ''b''1, ..., ''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 has height at most and all the numbers have height at most then the linear form : is either 0 or satisfies : where : and the field generated by and over the rationals has degree at most ''d''. In the special case when ''β''0 = 0 and all the 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 theBaker's method
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 are linearly independent over the rational numbers, for nonzero algebraic numbers then they are linearly independent over the algebraic numbers. 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 of several variables that vanishes to high order at many points of the form 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''.Construction of the auxiliary function
Assume there is a relation : for algebraic numbers ''α''1, ..., ''α''''n'', ''β''1, ..., ''β''''n''−1. The function Φ is of the form : 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 for integers with for some constant ''h''. This is possible because these conditions are homogeneous linear equations in the coefficients ''p'', which have a non-zero solution provided the number of unknown variables ''p'' is larger than the number of equations. The linear relation between the logs of the α's is needed to cut down the number of linear equations that have to be satisfied. Moreover, using Siegel's lemma, the sizes of the coefficients ''p'' can be chosen to be not too large. 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: *''L'' must be somewhat smaller than ''M'' to make the argument about extra zeros below work. *A small power of ''h'' must be larger than ''L'' to make the final step of the proof work. *''Ln'' must be larger than about ''M''''n''−1''h'' in order that it is possible to solve for the coefficients ''p''. 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 ''h''2 and ''L'' to be about ''h''2−1/2''n''. 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 ''M''''n''−1''h'' would become ''Ln'' must be larger than about ''Mnh'', which is incompatible with the condition that ''L'' is somewhat smaller than ''M''.Zeros of the auxiliary function
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. This is done by combining two ideas; First one shows that the derivatives at these points are quite small, by using the fact that many derivatives of Φ vanish at many nearby points. Then one shows that derivatives of Φ at this point are given by algebraic integers times known constants. If an algebraic integer has all its conjugates bounded by a known constant, then it cannot be too small unless it is zero, because the product of all conjugates of a nonzero algebraic integer is at least 1 in absolute value. 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 ''h''1+1/8''n'' consecutive integers 1, 2, 3, .... Repeating this argument ''J'' times shows that Φ vanishes to order ''M''/2''J'' at ''h''1+''J''/8''n'' points, provided that ''h'' is sufficiently large and ''L'' is somewhat smaller than ''M''/2''J''. One then takes ''J'' large enough that: : (''J'' larger than about 16''n'' will do if ''h''2 > ''L'') so that: :Completion of the proof
By definition 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. 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: : so are multiplicatively dependent. Taking logs shows that are linearly dependent over the rationals.Extensions and generalizations
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 2π''i'' in the theorem. This requires a modification of the final step of the proof. One shows that many derivatives of the function vanish at ''z'' = 0, by an argument similar to the one above. 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 . So two of these expressions must be the same which shows that log ''α''1,...,log ''α''''n'' are linearly dependent over the rationals. gave an inhomogeneous version of the theorem, showing that : is nonzero for nonzero algebraic numbers ''β''0, ..., ''βn'', ''α''1, ..., ''α''''n'', and moreover giving an effective lower bound for it. The proof is similar to the homogeneous case: one can assume that : and one inserts an extra variable ''z''0 into Φ as follows: :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 : where ''β''1 and ''β''2 are algebraic and ''λ''1 and ''λ''2 are in .See and for details. 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 ''λ''1, ..., ''λn'' are elements of 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. 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 outSee also
* Analytic subgroup theoremNotes
References
* * * * * *. * * * * * {{refend Transcendental numbers Theorems in number theory