2008/04/22 by Chuangxun Cheng, Cheng, Chuangxun, Brian C. Dietel +15
Mathematics · #11J81 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry and Number Theory #FOS: Mathematics #Meromorphic and Entire Functions #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.0804.3550
openalex publication_date 2008/04/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
During the Arizona Winter School 2008 (held in Tucson, AZ) we worked on the following problems: a) (Expanding a remark by S. Lang). Define E0 = ℚ Inductively, for n ≥ 1, define En as the algebraic closure of the field generated over En-1 by the numbers exp(x)=ex, where x ranges over En-1. Let E be the union of En, n ≥ 0. Show that Schanuel's Conjecture implies that the numbers π, log π, log log π, log log log π, … are algebraically independent over E. b) Try to get a (conjectural) generalization involving the field L defined as follows. Define L0 = ℚ. Inductively, for n ≥ 1, define Ln as the algebraic closure of the field generated over Ln-1 by the numbers y, where y ranges over the set of complex numbers such that ey∈ Ln-1. Let L be the union of Ln, n ≥ 0. We were able to prove that Schanuel's Conjecture implies E and L are linearly disjoint over ℚ.