Auxiliary polynomials for transcendence results

Abstract

The main goal of this work is to prove several transcendence results using auxiliary functions, and in doing so showcase their effectiveness in various contexts. The main theorems covered will be Hermite-Lindemann, Gelfond-Schneider, Schneider-Lang, and Baker’s theorem. We will employ two different proof strategies with auxiliary polynomials: two similar ones for Hermite-Lindemann and Schneider-Lang, and a noticeably different one for Baker’s theorem. Gelfond-Schneider will come as a corollary to Schneider-Lang. We will ease into these theorems however, by first delving into the preliminary results and background knowledge requiered to understand their proofs. This includes but is not limited to derivations over number fields, valuation theory and height functions, and complex analysis. Furthermore, we will take a detour into ellipitic functions after proving the Schneider-Lang theorem due to independent interest, and to present a few applications of the Schneider-Lang theorem, as it is the most general one we will present.