Complete Resolution of B.Shapiro's Conjecture 12
Lande Ma, Zhaokun Ma
TL;DR
For a real polynomial $p(x)$ of even degree $n$, Shapiro's Conjecture 12 asserts that $\sharp_r\Delta + \sharp_r p > 0$ where $\Delta=(n-1)(p'(x))^{2}-np(x)p''(x)$. The paper introduces a root-locus framework via the real rational function $RF(s)$, its gain $K(s)$, and the quotient $PP=\frac{p''(x)p(x)}{(p'(x))^{2}}$, linking real zeros of $\Delta$ to the topology of root loci and breakaway points. It proves a rigorous necessary-and-sufficient condition for real critical points and then uses detailed case analysis of $PP$-space to obtain a complete resolution: Shapiro's Conjecture 12 holds in nine mutually exclusive conditions and fails in four, providing a full classification of even-degree real polynomials in this context. The results showcase a novel application of root-locus methods from control theory to a pure-m mathematics problem, delivering precise criteria and a complete taxonomy with potential implications for related real-root questions.
Abstract
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 $(n-1)(p')^2 - np p''$ 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.