Involutions on Zilber fields

Submitted by Vincenzo Mantova on Tue, 12/07/2011 - 23:23


Logic Colloquium 2011




Tuesday, 12 July, 2011

In [1], Zilber defined the `pseudoexponentiation' as a well behaved alternative to complex exponentiation, and proved the categoricity of its axioms; he then conjectured that the complex exponentiation is exactly the model of cardinality of the continuum. This conjecture includes the long-standing Schanuel's Conjecture, and its solution is probably very far. The field with pseudoexponentiation of cardinality of the continuum is now commonly called `Zilber field'.

Motivated by the conjecture, much work has been done recently towards proving that the Zilber field and the complex field are at least similar, either by proving classical statements in the context of pseudoexponentiation, or doing the converse (generally assuming Schanuel's Conjecture).

In our work, we managed to prove a strong result in this direction: the pseudoexponentiation has a `pseudoconjugation', i.e., an involution of the field commuting with the exponential function in a similar way to complex conjugation, and its fixed field is exactly the field of real numbers.


[1] Boris Zilber, Pseudo-exponentiation on algebraically closed fields of characteristic zero, Annals of Pure and Applied Logic, vol. 132 (2005), no. 1, pp. 67–95.