We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
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.
