Midpoints and critical points

2025/12/08 by Gati, Yousra, Kostov, Vladimir Petrov
Mathematics · #Advanced Differential Equations and Dynamical Systems #Mathematical functions and polynomials #Meromorphic and Entire Functions

paper · doi:10.48550/arxiv.2512.07322

Abstract

For a degree 5 real polynomial with roots x₁≤ ⋯ ≤ x₅ and roots ξ₁≤ ⋯ ≤ ξ₄ of its derivative, we set z_j:=(x_j+x_j+1)/2, 1≤ j≤ 4. We prove that one cannot have at the same time min_{1≤ j≤ 3}(z_j+1-z_j)≥ min_{1≤ j≤ 3}(ξ_j+1-ξ_j) and max_{1≤ j≤ 3}(z_j+1-z_j)≥ max_{1≤ j≤ 3}(ξ_j+1-ξ_j). The result settles a general question about midpoints and critical points of hyperbolic polynomials.

Citations

• Numerical approach for solving problems arising from polynomial analysis
• A few riddles behind Rolle's theorem

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