On Crofoot-Sarason's conjecture for harmonic polynomials
The Crofoot-Sarason Conjecture states that for \(n\geq 2\) there exist \(n-1\) points \(z_1,\ldots,z_{n-1}\) and a polynomial \(p\) of degree \(n\) such that \(p(z_j)+\overline{z}_j=0\) and \(p'(z_j)=0\) for all \(j\). This conjecture is known to imply Wilmhurst's conjecture stating that \(p_n(z)+\overline{z}\) has at most \(3n-2\) zeros if \(p_n(z)\) is an analytic polynomial of degree \(n\). As stated in a remark added at the end of the article under review, L. Geyer has apparently confirmed the first conjecture (and hence the second one) for all integers in a yet unpublished manuscript. In the main body of the article, the authors give explicit arguments to prove the Crofoot-Sarason Conjecture true in the cases \(n=4,5,6,8\) (and state that their work was done before they learned of Geyer's result).
scientific article; zbMATH DE number 2104112