Schanuel's conjecture

In mathematics, specifically transcendence theory, Schanuel's conjecture is the following statement::Given any "n" complex numbers "z"₁,...,"z"_"n" which are linearly independent over the rational numbers Q, the extension field Q("z"₁,...,"z"_"n",exp("z"₁),...,exp("z"_"n")) has transcendence degree of at least "n" over Q.

The conjecture was formulated by Stephen Schanuel in the early 1960s and can be found in (Lang 1966) [Serge Lang. "Introduction to Transcendental Numbers." Addison-Wesley 1966. Pages 30-31] . No proof is known.

The conjecture, if proven, would imply the Lindemann-Weierstrass theorem, the Gelfond-Schneider theorem and several other results about transcendence properties of the exponential function, as well as the (as yet unproven) algebraic independence of π and "e".

The converse Schanuel conjecture [Scott W. Williams. [http://www.math.buffalo.edu/~sww/0papers/million.buck.problems.mi.pdf Million Bucks Problems] ] is the following statement: :Suppose "F" is a countable field with characteristic 0, and "e" : "F" → "F" is a homomorphism from the additive group ("F",+) to the multiplicative group ("F",·) whose kernel is cyclic. Suppose further that for any "n" elements "x"₁,...,"x"_"n" of "F" which are linearly independent over Q, the extension field Q("x"₁,...,"x"_"n","e"("x"₁),...,"e"("x"_"n")) has transcendence degree at least "n" over Q. Then there exists a field homomorphism "h" : "F" → C such that "h"("e"("x"))=exp("h"("x")) for all "x" in "F".

A version of Schanuel's conjecture for formal power series, also by Schanuel, was proven by James Ax in 1971. [James Ax. On Schanuel's conjectures. "Annals of Mathematics"(2) 93, 1971, pages 252-268.] It states::Given any "n" formal power series "f"₁,...,"f"_"n" in tC"t" which are linearly independent over Q, then the field extension C("t","f"₁,...,"f"_"n",exp("f"₁),...,exp("f"_"n")) has transcendence degree at least "n" over C("t").

Boris Zilber [Boris Zilber. Pseudo-exponentiation on algebraically closed fields of characteristic zero, Annals of Pure and Applied Logic, (132), 2004, 1, pp 67-95] constructed an axiomatization of pseudo-exponentiation in algebraically closed fields of characteristic zero. Using Hrushovski's construction, he proved that the theory is satisfiable, and categorical in all uncountable powers. The resulting theory has a unique model of cardinality of the continuum. If Schanuel's conjecture is true, then (C,+,x,exp) is the unique model of this cardinality. Conversely, the Hrushovski inequality formulated in these models is Schanuel's conjecture. This doesn't prove Schanuel's conjecture, however, since we don't know that the unique model is (C,+,x,exp).

ee also

*Schanuel's conjecture implies a positive solution to Tarski's exponential function problem

References