Transcendence theory
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In mathematics, transcendence theory is a branch of number theory that investigates transcendental numbers, in both qualitative and quantitative ways.
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[edit] Transcendence
The fundamental theorem of algebra tells us that if we have a non-zero polynomial with integer coefficients then that polynomial will have a root in the complex numbers. That is, for any polynomial P with integer coefficients there will be a complex number α such that P(α) = 0. Transcendence theory is concerned with the converse question, given a complex number α, is there a polynomial P with integer coefficients such that P(α) = 0? If no such polynomial exists then the number is called transcendental.
More generally the theory deals with algebraic independence of numbers. A set of numbers {α1,α2,…,αn} is called algebraically independent over a field k if there is no non-zero polynomial P in n variables with coefficients in k such that P(α1,α2,…,αn) = 0. So working out if a given number is transcendental is really a special case of algebraic independence where our set consists of just one number.
[edit] History
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[edit] Approaches
A typical problem in this area of mathematics is to work out whether a given number is transcendental. However this problem is in general extremely difficult. Cantor used a cardinality argument to show that there are only countably many algebraic numbers, and hence almost all numbers are transcendental. But despite this it is extremely difficult to actually prove that a given number is transcendental.
For this reason transcendence theory often works towards a more quantitative approach. So given a particular complex number α one can ask how close α is to being an algebraic number. For example, if one supposes that the number α is algebraic then can one show that it must have very high degree or a minimum polynomial with very large coefficients? Ultimately if it is possible to show that no finite degree or size of coefficient is sufficient then the number must be transcendental. Since a number α is transcendental if and only if P(α)≠0 for every non-zero polynomial P with integer coefficents, this problem can be approached by trying to find lower bounds of the form
- |P(α)| > F(A,d)
where the right hand side is some positive function depending on the some measure A of the size of the coefficients of P, and its degree d, and such that these lower bounds apply to all P ≠ 0. Such a bound is called a transcendence measure.
The case of d = 1 is that of "classical" diophantine approximation asking for lower bounds for
- |ax + b|.
The methods of transcendence theory and diophantine approximation have much in common: they both use the auxiliary function concept.
[edit] Major Results
The Gelfond-Schneider theorem was the major advance in transcendence theory in the period 1900-1950. In the 1960s the method of Alan Baker on linear forms in logarithms of algebraic numbers reanimated transcendence theory, with applications to numerous classical problems and diophantine equations.
[edit] Open Problems
While the Gelfond-Schneider theorem proved that a large class of numbers was transcendental, this class was still countable. Many well known mathematical constants are still not known to be transcendental, and in some cases it is not even known whether they are rational or irrational. A partial list can be found here.
A major problem in transcendence theory is showing that a particular set of numbers is algebraically independent rather than just showing that individual elements are transcendental. So while we know that e and π are transcendental that doesn't imply that e+π is transcendental, nor other combinations of the two. Another major problem is dealing with numbers that are not related to the exponential function. The main results in transcendence theory tend to revolve around e and the logarithm function, which means that wholly new methods tend to be required to deal with numbers that cannot be expressed in terms of these two objects in an elementary fashion.
Schanuel's conjecture would solve the first of these problems somewhat as it deals with algebraic independence and would indeed confirm that e+π is transcendental. It still revolves around the exponential function however and so would not necessarily deal with numbers such as Apéry's constant or the Euler–Mascheroni constant. Another extremely difficult unsolved problem is the so-called Constant or Identity problem[1].
[edit] References
- ^ Richardson, D. "Some Unsolvable Problems Involving Elementary Functions of a Real Variable." J. Symbolic Logic 33, pp.514-520, 1968.
- A.O. Gelfond, Transcendental and Algebraic Numbers, Dover Publications Inc., 1960.
- Serge Lang, Introduction to Transcendental Numbers, Addison-Wesley Publishing Company, 1966.
- Alan Baker, Transcendental Number Theory, Cambridge University Press, 1975.
