2014/02/28 by Vincenzo Mantova, V. Mantova, Umberto Zannier +1 · 10 citations
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Algebraic equation #Algebraic number #Commutative Algebra and Its Applications #Conjecture #Diophantine equation #Polynomial and algebraic computation #Transcendental equation #Transcendental number #Variable (mathematics) #math.LO #math.NT #msc:03C60 #msc:11D61
paper · pdf · doi:10.1112/blms/bdv096
published in Bulletin of the London Mathematical Society 48(2), 309-320 (Wiley) · 13 pages. Appendix by V. Mantova and U. Zannier. New title and various stylistic improvements
arxiv created 2015/11/24 · openalex publication_date 2016/02/22 · openalex created_date 2016/06/24 · arxiv updated 2017/02/01 · openalex updated_date 2026/08/05
Assuming Schanuel's conjecture, we prove that any polynomial–exponential equation in one variable must have a solution that is transcendental over a given finitely generated field. With the help of some recent results in Diophantine geometry, we obtain the result by proving (unconditionally) that certain polynomial–exponential equations have only finitely many rational solutions. This answers affirmatively a question of David Marker, who asked, and proved in the case of algebraic coefficients, whether at least the one variable case of Zilber's strong exponential-algebraic closedness conjecture can be reduced to Schanuel's conjecture.