Abstract

A tract (or asymptotic tract) of a real function u harmonic and nonconstant in the complex plane 𝒞 is one of the nc components of the set {z:u(z)≠c}, and the order of a tract is the number of non-homotopic curves from any given point to ∞ in the tract. The authors prove that if u(z) is an entire harmonic polynomial of degree n, if the critical points of any of its analytic completions f lie on the level sets τj={z:u(z)=cj}, where 1≤j≤p and p≤n−1, and if the total order of all the critical points of f on τj is denoted by σj, then {nc:c∈ℜ}={n+1}∪{n+1+σj:1≤j≤p}.