On the number of real zeroes of a homogeneous differential polynomial and a generalization of the Hawaii conjecture

For a given real polynomial p we study the possible number of real roots of a differential polynomial H_[p](x) = (p'(x))²-p(x)p''(x), ∈ R. In the special case when all real zeros of the polynomial p are simple, and all roots of its derivative $p'$ are real and simple, the distribution of zeros of H_[p] is completely described for each real . We also provide counterexamples to two Boris Shapiro's conjectures about the number of zeros of the function Hn-1/n[p].