Shapiro's Conjecture 12 on positivity of a zero-count sum for even-degree real polynomials

Prove that for any real polynomial p(x) of even degree n, the sum of the number of real zeros of the polynomial (n−1)(p′(x))^2 − n p(x) p′′(x) and the number of real zeros of p(x) is positive; that is, establish that #_r[(n−1)(p′)^2 − n p p′′] + #_r p > 0 for all such p, where #_r denotes the number of real zeros.

Background

The conjecture was posed by B. Shapiro (2015) in the context of refining classical results on the distribution of real zeros of polynomials and related differential polynomials. It asserts a universal positivity statement combining the real zero count of a specific differential polynomial with that of the original even-degree real polynomial.

In this paper, the authors develop a root-locus approach from control theory to analyze the rational function p''p/(p')^2 and classify when the resulting differential polynomial has real zeros. They claim a complete resolution by proving that the conjecture holds in nine mutually exclusive cases and fails in four, thereby providing a full classification of even-degree real polynomials relative to this assertion.