In mathematics, specifically transcendental number theory, Schanuel's conjecture is a conjecture about the transcendence degree of certain field extensions of the rational numbers
, which would establish the transcendence of a large class of numbers, for which this is currently unknown.
It is due to Stephen Schanuel and was published by Serge Lang in 1966.
Schanuel's conjecture, if proven, would generalize most known results in transcendental number theory and establish a large class of numbers transcendental.
The first proof for this more general result was given by Carl Weierstrass in 1885.
[4] This so-called Lindemann–Weierstrass theorem implies the transcendence of the numbers e and π.
It further gives the transcendence of the trigonometric functions at nonzero algebraic values.
Another special case was proved by Alan Baker in 1966: If complex numbers
are chosen to be linearly independent over the rational numbers
are also linearly independent over the algebraic numbers
Schanuel's conjecture would strengthen this result, implying that
[2] In 1934 it was proved by Aleksander Gelfond and Theodor Schneider that if
This establishes the transcendence of numbers like Hilbert's constant
[5] The Gelfond–Schneider theorem follows from Schanuel's conjecture by setting
It also would follow from the strengthened version of Baker's theorem above.
Schanuel's conjecture, if proved, would also establish many nontrivial combinations of e, π, algebraic numbers and elementary functions to be transcendental:[2][6][7] In particular it would follow that e and π are algebraically independent simply by setting
If Schanuel's conjecture is true then this is, in some precise sense involving exponential rings, the only non-trivial relation between e, π, and i over the complex numbers.
[10] It states: Although ostensibly a problem in number theory, Schanuel's conjecture has implications in model theory as well.
Angus Macintyre and Alex Wilkie, for example, proved that the theory of the real field with exponentiation,
exp, is decidable provided Schanuel's conjecture is true.
[11] In fact, to prove this result, they only needed the real version of the conjecture, which is as follows:[12] This would be a positive solution to Tarski's exponential function problem.
A related conjecture called the uniform real Schanuel's conjecture essentially says the same but puts a bound on the integers mi.
[12] Macintyre and Wilkie showed that a consequence of Schanuel's conjecture, which they dubbed the Weak Schanuel's conjecture, was equivalent to the decidability of
This conjecture states that there is a computable upper bound on the norm of non-singular solutions to systems of exponential polynomials; this is, non-obviously, a consequence of Schanuel's conjecture for the reals.
In this setting Grothendieck's period conjecture for an abelian variety A states that the transcendence degree of its period matrix is the same as the dimension of the associated Mumford–Tate group, and what is known by work of Pierre Deligne is that the dimension is an upper bound for the transcendence degree.
[13] While a proof of Schanuel's conjecture seems a long way off,[14] connections with model theory have prompted a surge of research on the conjecture.
In 2004, Boris Zilber systematically constructed exponential fields Kexp that are algebraically closed and of characteristic zero, and such that one of these fields exists for each uncountable cardinality.
[15] He axiomatised these fields and, using Hrushovski's construction and techniques inspired by work of Shelah on categoricity in infinitary logics, proved that this theory of "pseudo-exponentiation" has a unique model in each uncountable cardinal.
Schanuel's conjecture is part of this axiomatisation, and so the natural conjecture that the unique model of cardinality continuum is actually isomorphic to the complex exponential field implies Schanuel's conjecture.
[16] As this construction can also give models with counterexamples of Schanuel's conjecture, this method cannot prove Schanuel's conjecture.