Bialgebraic geometry of Böttcher coordinates

Abstract: The Böttcher coordinate $\Psi$ of a non-exceptional polynomial $f$ which is defined on a disk $D$ centered at the origin and satisfies

$\Psi(z^d) = f(\Psi(z))$

where $d = deg(f)$, is known to be transcendental. In this talk, I will discuss the associated $f$-bialgebraic sets, that are, algebraic branches in $D^n$ whose images under the coordinatewise map $(\Psi , ... , \Psi)$ are again algebraic branches of the same dimension. Under mild assumptions on the Julia set of $f$, we obtain a complete classification of these sets. I will also explain how this classification leads to an Ax–Lindemann–Weierstrass type conjecture, which we prove when the Julia set of $f$ is disconnected.

algebraic geometrynumber theory

Audience: researchers in the topic

Comments: Note this is NOT the usual room for this semester! This seminar is joint with the dynamics seminar, and will be in CDS 548.

Boston University Number Theory Seminar