Schanuel Nullstellensatz for Zilber fields

Abstract

We characterize the unsolvable exponential polynomials over the exponential fields introduced by Zilber, and deduce Picard's Little Theorem for such fields.

Model theory of special subvarieties and Schanuel-type conjectures

“…In [13] P. D'Aquino, A.Macintyre and G.Terzo find a context in which the Identity Theorem of complex analysis can be checked and confirmed in F exp .…”

“…Since F exp comes with no topology there is no direct counterpart to the notion of accumulation in it. Instead [13] considers specific subsets of F exp , such that Q or the torsion points where an accumulation point can be defined in the same way as in C. In such cases, for a certain type of functions f D'Aquino, Macintyre and Terzo prove that the conclusion of the Identity Theorem holds in F exp . It is interesting that the results are obtained by invoking deep theorems of Diophantine Geometry.…”

Finitely presented exponential fields

“…It is immediate that (4) implies (2), that (4) implies (5), and that (5) implies (6). We now assume that there is a finitec ⊳ F .…”

Comparing and Zilber’s Exponential Fields: Zero Sets of Exponential Polynomials

“…There are many novelties in his analysis, including a reinterpretation of Schanuel's Conjecture in terms of Hrushovski's very general theory of predimension and strong extensions. By now there is no need to spell out yet again all his ingredients and results (see [18], [6]). The most dramatic aspect is that his fields satisfy Schanuel's Conjecture and a beautiful Nullstellensatz for exponential equations.…”

“…We have undertaken a research programme of taking results from C, proved using analysis and/or topology, and seeking exponential-algebraic proofs in B. An early success was a proof in B of the Schanuel's Nullstellensatz [6], proved in C [13] using Nevanlinna theory. More recently, in [7], we derived, in an exponential-algebraic way, Shapiro's Conjecture from Schanuel's Conjecture, thereby getting Shapiro's Conjecture in B.…”