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ON THE NUMBER OF REAL ZEROS OF POLYNOMIALS OF EVEN DEGREE

Part of: Geometric function theory Polynomials, rational functions Real and complex fields

Published online by Cambridge University Press:  27 February 2023

LANDE MA Open the ORCID record for LANDE MA [Opens in a new window]  and
ZHAOKUN MA Open the ORCID record for ZHAOKUN MA [Opens in a new window]
LANDE MA*
Affiliation:
School of Mathematical Sciences, Tongji University, Shanghai 200092, PR China
ZHAOKUN MA
Affiliation:
YanZhou College, ShanDong Radio and TV University, YanZhou, ShanDong 272100, PR China e-mail: dzy200408@126.com
*
e-mail: dzy200408@sina.cn
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Abstract

For any real polynomial $p(x)$ of even degree k, Shapiro [‘Problems around polynomials: the good, the bad and the ugly$\ldots $’, Arnold Math. J. 1(1) (2015), 91–99] proposed the conjecture that the sum of the number of real zeros of the two polynomials $(k-1)(p{'}(x))^{2}-kp(x)p{"}(x)$ and $p(x)$ is larger than 0. We prove that the conjecture is true except in one case: when the polynomial $p(x)$ has no real zeros, the derivative polynomial $p{'}(x)$ has one real simple zero, that is, $p{'}(x)=C(x)(x-w)$, where $C(x)$ is a polynomial with $C(w)\ne 0$, and the polynomial $(k-1)(C(x))^2(x-w)^{2}-kp(x)C{'}(x)(x-w)-kC(x)p(x)$ has no real zeros.

Keywords

real zerospolynomialsRolle’s theoremconjecture of B. Shapiro

MSC classification

Primary: 26C10: Polynomials
Secondary: 12D10: Polynomials 30C10: Polynomials 30C15: Zeros of polynomials, rational functions, and other analytic functions (e.g. zeros of functions with bounded Dirichlet integral)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 109 , Issue 1 , February 2024 , pp. 89 - 93
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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References

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