Minimal polynomial degree for geometric separation of torus-knot subsets on the standard torus

Determine, for each integer p ≥ 1, the minimal integer degree d of a polynomial P(z,w) such that P(0,0) = 0 and the level sets H_λ = {(z,w) ∈ C^2 : P(z,w) = λ} are disjoint from K_p = {(e^{ipθ}, e^{-iθ}) : 0 ≤ θ ≤ 2π} for all λ ≥ 0, thereby showing that (0,0) does not belong to the geometric d-polynomial hull of K_p.

Background

In Section 2 the paper studies polynomial convexity with degree bounds for subsets of the standard Lagrangian torus in C^2. For the family K_p = {(e^{ipθ}, e^{-iθ})}, the author exhibits, for p ≤ 4, explicit quadratic polynomials whose level sets avoid K_p for all nonnegative levels while passing through (0,0), which demonstrates that (0,0) is not in the geometric degree-2 polynomial hull of K_p.

The author suggests this behavior may fail for larger p but leaves open the task of identifying the optimal (lowest) degree required to achieve such a separation for general p.