The arithmetic of power series - Lecture 2
Appears in collection : Diophantine approximation and transcendence 2025 / Approximation diophantienne et transcendance 2025
We explain how the classical Thue hypergeometric Pade method gets encoded into the formal framework of the previous talk by Yunqing Tang on our joint work with Frank Calegari, and how it continues with the more recent methodology of multivalent holonomy bounds. The primary focus here is on the algebraic Apery limits, and therefore on effectivity. We outline self-contained proof of a basic, but completely explicit holonomy bound that includes a Diophantine approximation term, and we explain how to apply it to effectivize the Thue-Siegel square root exponent for Diophantine approximation to high order roots from a fixed rational number. well-known geometry of numbers argument of Bombieri's allows to thereafter recover an effective height bound on the solutions to the general S-unit equation in two variables.
Document25.pdf : Complete lecture slides (with page 24/43 correcting a typo at time 14:30 of the recording).
