Complete Resolution of B.Shapiro's Conjecture 12

For any real polynomial $p(x)$ of even degree $n$, Shapiro [{\it Arnold Math. J.} 1(1) (2015), 91–99] conjectured that the sum of the number of real zeros of $(n-1)(p^{\mathrm{\prime }}{)}^2-np{p}^{\mathrm{\prime }\mathrm{\prime }}$ and the number of real zeros of $p$ is positive. We resolve this conjecture completely: it holds in nine mutually exclusive cases and fails in four, as characterized by the root locus properties of general real rational functions. Our results provide a complete classification of real polynomials of even degree with respect to this conjecture.