The exact bound on the number of zeros of harmonic polynomials
Let \(p(z)\), \(q(z)\) be complex polynomials of degree \(n\) and \(m<n\) respectively. Then it is known, that the harmonic polynomial \(p(z)+\overline{q(z)}\) has at most \(n^2\) zeros in \(\mathbb{C}\). Using a generalized argument principle, the authors construct for each \(n\in\mathbb{N}\) a harmonic polynomial \(f_n(z)\) of degree \(n\) having precisely \(n^2\) zeros. For \(n\) even (and similarly for \(n\)) odd \(f_{n+2}(z):=p_{n+2}(z)+ \overline{q_{n+2}(z)}\), where \[ p_{n+2}(z):= \int^z_0(b^3_n+u^2) b^{-3}_np_n'(u) du,\quad p_2(z):=z^2/2, \] \[ q_{n+2}(z):= \int^z_0(b^3_n+ u^2(1+b^2_n)) b^{-3}_nq_n'(u)du,\quad q_2(z):= z\quad\text{and where }b_2,b_4,\dots,b_n\text{ are }>0 \] and sufficiently large. Here, \(f_2(z)= z^2/2+\overline z\) has the zeros \(0\), \(-2\), \(1\pm i\sqrt 3\).
scientific article; zbMATH DE number 871562