Problems Around Polynomials — Conjecture 12

Definitions and mathematical statement

Let $p\in{\mathbb{R}}[x]$ have even degree, and let $p',p''$ denote its derivatives. The printed conjecture, attributed to B. Sh, is \[ \#_r\!\left[(k-1)(p')^2-kpp''\right]+\#_r p>0. \] The record does not define or quantify $k$. In particular it does not say whether $k=\deg p$, a fixed parameter, or an independently quantified variable. No such relation is silently imposed.

The notation $\#_r q$ evidently concerns real roots of a polynomial $q$, but the supplied record does not define its counting convention. For nonzero polynomials, either the distinct-root count or the count with multiplicity is nonnegative, and positivity of their sum means that at least one of \[ p(x),\qquad (k-1)(p'(x))^2-kp(x)p''(x) \] has a real zero. This explains the strict inequality without selecting a multiplicity convention. The second expression can be identically zero for some choices; the root-count convention for that case is not supplied. The source also does not exclude constant polynomials, which have even degree zero, or otherwise state a minimum degree.

Accordingly, the exact formula is retained, but a fully quantified numerical conjecture requires recovering $k$, the degree range, and the zero-polynomial root-count convention from the original context.

Short English statement

The conjecture asserts that an even-degree real polynomial or the displayed derivative combination must have a real root. The supplied assessment labels the conjecture false: it attributes counterexamples in every even degree at least four to Katkova--Tyaglov--Vishnyakova and a 2025 classification to Ma--Ma, with nine disjoint cases where the inequality holds and four where it fails. The record supplies neither the cases nor complete references, and leaves the parameter $k$ undefined, so those resolution claims are retained as attributed rather than asserted as a verified theorem for a guessed formulation.

Sources

• UnsolvedMath permanent ID 2200019, source code catalogue entry, "Problems Around Polynomials --- Conjecture 12." Registry curation and metadata: Ulam AI and UnsolvedMath Contributors, CC BY 4.0; updated 2026-08-06.
• Shapiro, Problems Around Polynomials: The Good, The Bad and The Ugly (2015), source-order item 19, Conjecture 12 as printed, attribution B. Sh: https://arxiv.org/abs/1503.05295. Supplied source-TeX extraction accessed 2026-07-29. Original metadata presents an open item; status/release and rights review are NEEDS_REVIEW.
• Katkova--Tyaglov--Vishnyakova and Ma--Ma (2025), as named in the supplied resolving assessment. No titles, venues, locators, counterexample formulas, or classification cases are supplied; none are invented.