Brenti’s problem on palindromic real-rooted polynomials and h-vectors of polytopes

For every polynomial $f(t)\in\mathbb{Z}[t]$ that is monic, has nonnegative coefficients, is real-rooted, and is palindromic in the sense that $f(t)=t^d f(t^{-1})$ for $d=\deg f$ , there exists a simplicial convex polytope $P$ such that its $h$ -polynomial satisfies $h_P(t)=f(t)$ .

References

Primary source

arXiv

Additional references

Palindromic real-rooted polynomials and h-vectors of polytopes — arXiv

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

A new unverified preprint claims to settle Brenti’s long-standing question affirmatively, but no independent confirmation was found.

Brenti’s problem, circulated since 2004, concerns a proposed connection between palindromic real-rooted polynomials and the polynomial data of simplicial convex polytopes.

September 2026 claimed affirmative answer

An unrefereed arXiv preprint claims an affirmative solution to the problem. The retrieved evidence gives no independent verification, referee report, or discussion identifying a gap.

Current status (as of September 2026): An arXiv preprint claims the problem is solved affirmatively, but the claim remains unverified; no contrary result or independently confirmed proof was found.

Sources

No solutions have been posted yet.