Polynomial estimates over exponential curves in $\mathbb C^2$

TL;DR

This paper extends Bernstein-Walsh inequalities to exponential curves in complex space for non-real parameters, broadening the scope of previous results limited to real parameters.

Contribution

It generalizes Bernstein-Walsh inequalities to complex exponential curves, expanding prior work that was restricted to real parameters.

Findings

Bernstein-Walsh inequality holds for complex exponential curves with non-zero imaginary part

Extension of Coman-Poletsky's theorem from real to complex parameters

Provides new estimates on polynomial behavior over exponential curves in a7^2

Abstract

For any complex $\alpha$ with non-zero imaginary part we show that Bernstein-Walsh type inequality holds on the piece of the curve $\{(e^z,e^{\alpha z}) : z \in \mathbb C\}$. Our result extends a theorem of Coman-Poletsky \cite{CP10} where they considered real-valued $\alpha$.