Speaker: Damien Roy
23 September, 2020 13:30 UTC
Abstract: The theorem of Lindemann-Weierstrass asserts that the exponentials of distinct algebraic numbers are linearly independent over the field of rational numbers. The proof uses a construction of simultaneous rational approximations to such exponentials values, which goes back to Hermite. In this talk, we show that, from an adelic perspective, these approximations are essentially best possible. This point of view partly explains the nature of the algebraic numbers whose exponentials have a structured continuous fraction expansion. We also propose few specific conjectures regarding simultaneous approximations to such values in adèle rings.
The proof of our main result requires a separate analysis for each place of the associated number field. For the Archimedean places, it relies on the structure of the graph drawn in the complex plane by the paths of fastest descent for the norm of a general univariate complex polynomial starting from the roots of its derivative and ending in the roots of the polynomial. It happens that this graph is a tree and that the lengths of those paths can be estimated from above in terms of the degree of the given polynomial and the diameter of its set of zeros. We will mention some instances where these upper bounds can be greatly improved and state as an open problem whether or not such improvements hold in general.