Degree-Bounded Polynomial Convexity of Circular and Smooth Arcs

We prove that the circular arc $A_{\alpha }=\{e^{it};\mathrm{\mid }t\mathrm{\mid }\le \alpha \}$ is $d$-polynomially convex if and only if $\alpha \le \frac{d-1}{d}\pi \mathrm{.}$ We then study d-polynomial convexity of smooth Jordan arcs in terms of their total absolute curvature $T(K)\mathrm{.}$ 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: $T(K)\le \pi \mathrm{/}2$ implies that $K$ is $2$-polynomially convex.