Asymptotic tracts of harmonic functions III

A tract (or asymptotic tract ) of a real function u harmonic and nonconstant in the complex plane 𝒞 is one of the n c 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 τ j = { z : u ( z ) = c j } , where 1 ≤ j ≤ p and p ≤ n − 1 , and if the total order of all the critical points of f on τ j is denoted by σ j , then { n c : c ∈ ℜ } = { n + 1 } ∪ { n + 1 + σ j : 1 ≤ j ≤ p } .