In mathematics, transcendence theory investigates transcendental numbers, in a qualitative and quantitative way.

The qualitative approach is concerned with a given number, such as e. It was proved in the nineteenth century that e is indeed transcendental, i.e. that for any integral polynomial P(x) such that

P(e) = 0

we must have that P is the zero polynomial.

The quantitative approach asks one to find lower bounds

P(e) > F(A,d)

depending on a bound A of the coefficients of P and its degree, that apply to all P ≠ 0. Such a bound is called a transcendence measure.

The case of d = 1 is closely related to diophantine approximation theory, in that it asks for lower bounds for

|ae + b|,

and this is essentially the same problem as lower bounds for

|e + b/a|,

i.e. the approximation of e by rational numbers, with bounded numerator and denominator. The methods of transcendence theory and diophantine approximation have much in common: they both use the auxiliary function concept.

More generally transcendence theory deals with the algebraic independence of sets of numbers. This corresponds to taking P above to be a polynomial in several variables, considering P(x,y, ...) evaluated at given fixed values, as P varies. There are some standard conjectures, for example Schanuel's conjecture, that describe the expected algebraic independence of 'classical numbers', such as e and π. Other numbers, such as periods of abelian integrals, are interesting examples for transcendence theory.

The Gel'fond-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.