On the Number of Real Zeros of Polynomials of Even Degree

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.

The domains where Smale's mean value problem does not hold

“…In our previous work [12], we comprehensively solved the 12th conjecture proposed by B. Shapiro, which inspired us to address Smale's mean value conjecture. However, the domains where the Smale's mean value conjecture doesn't hold haven't been given.…”