The Newman algorithm for constructing polynomials with restricted coefficients and many real roots

2024/04/11 by Markus Jacob, Fëdor Nazarov, Jacob, Markus +1
Engineering · Mathematics · #Advanced Numerical Analysis Techniques #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Dynamics and Fractals #Mathematical functions and polynomials

paper · pdf · doi:10.48550/arxiv.2404.07971

openalex publication_date 2024/04/11 · openalex created_date 2024/04/13 · openalex updated_date 2026/07/28

Abstract

Under certain natural sufficient conditions on the sequence of uniformly bounded closed sets E_k⊂ℝ of admissible coefficients, we construct a polynomial P_n(x)=1+∑_k=1ⁿε_k x^k, ε_k∈ E_k, with at least c√(n) distinct roots in [0,1], which matches the classical upper bound up to the value of the constant c>0. Our sufficient conditions cover the Littlewood (E_k=\-1,1\) and Newman (E_k=\0,(-1)^k\) polynomials and are also necessary for the existence of such polynomials with arbitrarily many roots in the case when the sequence E_k is periodic.

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