Degree-bounded polynomial convexity of circular and smooth arcs

Let d≥ 1 be an integer. We study d-polynomial convexity of smooth Jordan arcs in terms of their total absolute curvature (K). We prove that every ²-smooth Jordan arc K⊂ satisfying (K)≤ d-1/dπ is d-polynomially convex. This bound is sharp: for every τ>d-1/dπ, there exists a smooth Jordan arc K⊂ such that (K)<τ and K is not d-polynomially convex. We also show that, for $0<α<π$, the circular arc A_α=ⁱᵗ:|t|≤ α\ is d-polynomially convex if and only if α≤ d-1/dπ.