Problems Around Polynomials — Conjecture 11

Definitions and mathematical statement

For a nonzero real polynomial $q$, let $\#_r(q)$ denote the number of its real zeros and $\#_{nr}(q)$ the number of its nonreal complex zeros, counted with algebraic multiplicity in the convention used here. The supplied record does not explicitly fix multiplicity counting. Interpret a real polynomial with simple real zeros as requiring each of its real zeros to have multiplicity one, while allowing nonreal zeros; it does not mean that all its zeros are real.

Conjecture (B. Sh. in the source). For every real polynomial $p\in\mathbb R[x]$ of degree $k$ with simple real zeros, is \[ \#_r\big((k-1)(p')^2-kpp''\big)\leq\#_{nr}(p)? \] For a nondegenerate root-counting formulation take $k\geq2$; the source does not state a lower bound, and for $k=1$ the polynomial on the left is identically zero, for which a finite zero count is not defined. The same degeneracy occurs for constant input. Thus these cases require a convention or exclusion.

The source calls this the Hawaiian conjecture for $G_1$; the displayed polynomial, rather than an unspecified definition of $G_1$, is the target here. The supplied research summary reports the conjecture false and attributes counterexamples to Katkova, Tyaglov, and Vishnyakova. It provides neither a counterexample nor an identifiable publication, so the negative resolution is recorded as a literature attribution rather than independently established here.

Short English statement

Compare the number of real roots of $(k-1)(p')^2-kpp''$ with the number of nonreal roots of $p$. The supplied summary reports counterexamples, while the degree and root-counting conventions need clarification.

Sources

• B. Shapiro, Problems Around Polynomials: The Good, The Bad and The Ugly (2015), source-order item 18, printed Conjecture 11, https://arxiv.org/abs/1503.05295; source-TeX extraction, accessed 2026-07-29. The statement labels the attribution B. Sh. and includes the assessment seems good.
• Katkova, Tyaglov, and Vishnyakova, counterexamples to the Hawaiian-for-$G_1$ inequality, as named by the supplied research summary. No title, date, theorem number, or publication identifier is supplied.
• Local registry data/unsolvedmath/problems.json, permanent ID $2200018$, source code catalogue entry, Problems Around Polynomials --- Conjecture 11. The record labels this solved and SOLVED-IN-LITERATURE, with the conjecture reported false; the source-list background retains NEEDS_REVIEW status and rights qualifications.