Schanuel's Conjecture - Related Conjectures and Results

Related Conjectures and Results

The converse Schanuel conjecture 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 x₁,...,x_n of F which are linearly independent over Q, the extension field Q(x₁,...,x_n,e(x₁),...,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. It states:

Given any n formal power series f₁,...,f_n in tC] which are linearly independent over Q, then the field extension C(t,f₁,...,f_n,exp(f₁),...,exp(f_n)) has transcendence degree at least n over C(t).

As stated above, the decidability of R_exp follows from the real version of Schanuel's conjecture which is as follows:

Suppose x₁,...,x_n are real numbers and the transcendence degree of the field Q(x₁,...,x_n, exp(x₁),...,exp(x_n)) is strictly less than n, then there are integers m₁,...,m_n, not all zero, such that m₁x₁ +...+ m_nx_n = 0.

A related conjecture called the uniform real Schanuel's conjecture essentially says the same but puts a bound on the integers m_i. The uniform real version of the conjecture is equivalent to the standard real version. 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 R_exp. 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.

It is also known that Schanuel's conjecture would be a consequence of conjectural results in the theory of motives. There 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. Bertolin has shown how a generalised period conjecture includes Schanuel's conjecture.