Schanuel Property SC_K

Establish the Schanuel property SC_K: Let K be a subfield of C of finite transcendence degree; show that for all tuples x ⊆ C, the inequality trd(e^x) + ld(x/K) − ld(x) ≥ −trd(K) holds.

Background

The paper uses a Schanuel-type property to obtain a lower bound for the predimension function needed in the Hrushovski construction over real closed fields. Specifically, the authors assume SC_K for K = Q(βi) to locate a finitely rcl-generated substructure B ensuring (\bar{R}, G) lies in the class C_B, which is essential for proving that the real field with a dense family of logarithmic spirals is a model of their theory.

While a special case is known (SC_K holds for K = Q(βi) when β is exponentially transcendental, by Bays–Kirby–Wilkie), the general SC_K remains a conjectural statement, and the paper explicitly frames it as a conjecture to support the construction.

References

The Schanuel Conjecture, if proved, would settle many well-known problems in transcendental number theory. Its proof is, however, beyond the scope of this paper. The following weaker form of the conjecture is used in the proof of Lemma \ref{le:lb}. Let K be a subfield of C of finite transcendence degree. Then for all x\subseteqC, we have \trd(e{x})+\ld(x/K)-\ld(x)\geq-\trd(K).

— Green points in the reals  (2501.01176 - Zhang, 2 Jan 2025) in Section 2.2 (Facts in algebra), Conjecture [SC_K]

Determining the exact value of trdeg_{Q}Q(\pi,e)\in{1,2} remains open; it would equal 2 under Schanuel's conjecture.

— On Finite Gaussian Mixtures: Finiteness of the Number of Modes and an Application to NPMLE  (2608.16675 - Wang, 17 Aug 2026) in Remark following the definition of transcendence degree, Section 2, subsection “Preliminaries on Ax--Schanuel functional-transcendence theory”