Transcendental Function - Exceptional Set

Exceptional Set

If ƒ(z) is an algebraic function and α is an algebraic number then ƒ(α) will also be an algebraic number. The converse is not true: there are entire transcendental functions ƒ(z) such that ƒ(α) is an algebraic number for any algebraic α. In many instances, however, the set of algebraic numbers α where ƒ(α) is algebraic is fairly small. For example, if ƒ is the exponential function, ƒ(z) = ez, then the only algebraic number α where ƒ(α) is also algebraic is α = 0, where ƒ(α) = 1. For a given transcendental function this set of algebraic numbers giving algebraic results is called the exceptional set of the function, that is the set

If this set can be calculated then it can often lead to results in transcendence theory. For example, Lindemann proved in 1882 that the exceptional set of the exponential function is just {0}. In particular exp(1) = e is transcendental. Also, since exp(iπ) = -1 is algebraic we know that iπ cannot be algebraic. Since i is algebraic this implies that π is a transcendental number.

In general, finding the exceptional set of a function is a difficult problem, but it has been calculated for some functions:

• ,

• ,
  • Here j is Klein's j-invariant, H is the upper half-plane, and is the degree of the number field Q(α). This result is due to Theodor Schneider.

• ,
  • This result is a corollary of the Gelfond–Schneider theorem which says that if α is algebraic and not 0 or 1, and if β is algebraic and irrational then αβ is transcendental. Thus the function 2x could be replaced by cx for any algebraic c not equal to 0 or 1. Indeed, we have:

• A consequence of Schanuel's conjecture in transcendental number theory would be that

• A function with empty exceptional set that doesn't require one to assume this conjecture is the function ƒ(x) = exp(1 + πx).

While calculating the exceptional set for a given function is not easy, it is known that given any subset of the algebraic numbers, say A, there is a transcendental function ƒ whose exceptional set is A. Since, as mentioned above, this includes taking A to be the whole set of algebraic numbers, there is no way to determine if a function is transcendental just by looking at its values at algebraic numbers. In fact, Alex Wilkie showed that the situation is even worse: he constructed a transcendental function ƒ: R → R that is analytic everywhere but whose transcendence cannot be detected by any first-order method.