On the number of real zeroes of a homogeneous differential polynomial and a generalization of the Hawaii conjecture

2024/06/02 by Olga Katkova, Katkova, Olga, Mikhail Tyaglov +3
Computer Science · Mathematics · #26C10 #30C15 #Advanced Differential Equations and Dynamical Systems #Complex Variables (math.CV) #FOS: Mathematics #Meromorphic and Entire Functions #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2406.00686

openalex publication_date 2024/06/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For a given real polynomial p we study the possible number of real roots of a differential polynomial H_\varkappa[p](x) = \varkappa(p'(x))²-p(x)p''(x), \varkappa ∈ ℝ. In the special case when all real zeros of the polynomial p are simple, and all roots of its derivative p' are real and simple, the distribution of zeros of H_\varkappa[p] is completely described for each real \varkappa. We also provide counterexamples to two Boris Shapiro's conjectures about the number of zeros of the function H_(n-1)/(n)[p].

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