On the Koebe Quarter Theorem for Polynomials

2019/04/24 by Jimmy Dillies, Dillies, Jimmy, Dmitriy Dmitrishin +5 · 1 citation
Computer Science · Mathematics · #30C10 #30C25 (Primary) #30C55 #30C75 (Secondary) #Advanced Differential Equations and Dynamical Systems #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #FOS: Mathematics #Mathematics and Applications #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.1904.11039

openalex publication_date 2019/04/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

D. Dimitrov has posed the problem of finding polynomials that set the sharpness of the Koebe Quarter Theorem for polynomials and asked whether Suffridge polynomials are optimal. We disprove Dimitrov's conjecture for polynomials of degree 3, 4, 5 and 6. For polynomials of degree 1 and 2 the conjecture is obviously true. On the way we introduce a new family of polynomials that allows us to state a conjecture about the value of the Koebe radius for polynomials of a specific degree.

Cited by

• On a conjecture about univalent polynomials

Related

• Dimitrov's question for the polynomials of degree 1,2,3,4,5,6
• On Suffridge polynomials
• Koebe's theorem for trinomials with fold symmetry
• On a conjecture about univalent polynomials
• On C. Michel's hypothesis about the modulus of typically real polynomials