Degree-Bounded Polynomial Convexity of Circular and Smooth Arcs

Abstract

We prove that the circular arc $A_\alpha=\{e^{it};|t|\le\alpha\}$ is $d$-polynomially convex if and only if $\alpha\le \frac{d-1}{d}\pi.$ We then study d-polynomial convexity of smooth Jordan arcs in terms of their total absolute curvature $T(K).$ For every $\tau>\frac{d-1}{d}\pi$, we construct a smooth Jordan arc $K$ with $T(K)<\tau$ that is not $d$-polynomially convex. For $d=2$, this threshold is sharp: $T(K)\le\pi/2$ implies that $K$ is $2$-polynomially convex.