Degree-Bounded Polynomial Convexity of Circular and Smooth Arcs

Abstract

Let $d\ge 1$ be an integer. We study $d$-polynomial convexity of smooth Jordan arcs in terms of their total absolute curvature $\T(K)$. We prove that every $\cC^2$-smooth Jordan arc $K\subset\C$ satisfying $\T(K)\le \frac{d-1}{d}\pi$ is $d$-polynomially convex. This bound is sharp: for every $\tau>\frac{d-1}{d}\pi$, there exists a smooth Jordan arc $K\subset\C$ such that $\T(K)<\tau$ and $K$ is not $d$-polynomially convex. We also show that, for $0<\alpha<\pi$, the circular arc $A_\alpha=\{e^{it}:|t|\le \alpha\}$ is $d$-polynomially convex if and only if $\alpha\le \frac{d-1}{d}\pi$.