For a nonzero real polynomial r of degree d, define W(r) by ∑_n≥ 0 r(n)t^n=fracW(r)(t)(1-t)^d+1. If the coefficients of both W(p) and W(q) are nonnegative and log-concave, with no internal zeros, then the coefficients of W(pq) are also nonnegative and log-concave, with no internal zeros. Here a finite sequence (a_i) is log-concave if a_i^2≥ a_i-1a_i+1 whenever the terms are defined, and it has no internal zeros if its nonzero entries form an interval.

Preservation of log-concavity under Hadamard products

For a nonzero real polynomial $r$ of degree $d$ , define $\mathcal{W}(r)$ by $\sum_{n\ge 0} r(n)t^n=\frac{\mathcal{W}(r)(t)}{(1-t)^{d+1}}$ . If the coefficients of both $\mathcal{W}(p)$ and $\mathcal{W}(q)$ are nonnegative and log-concave, with no internal zeros, then the coefficients of $\mathcal{W}(pq)$ are also nonnegative and log-concave, with no internal zeros. Here a finite sequence $(a_i)$ is log-concave if $a_i^2\ge a_{i-1}a_{i+1}$ whenever the terms are defined, and it has no internal zeros if its nonzero entries form an interval.

References

Primary source

arXiv

Additional references

Preservation of log-concavity under Hadamard products — arXiv — Yanxin Liu, Jianxi Mao

algebraic-combinatorics polynomial-theory

Posted 3 weeks ago · edited 3 weeks ago

Progress summary

Refreshed 3 weeks ago

Claimed solved

An unrefereed preprint claims to prove the conjectured closure rule and extend it to repeated products and products of lattice polytopes.

The conjecture asks whether log-concavity survives coefficientwise, or Hadamard, products. The reported result also treats finite products and Cartesian products of lattice polytopes.

Known results

Wagner, 1992: Hadamard products preserve the Pólya-frequency property for polynomially interpolated sequences.
A 2024 preprint proves related preservation results for ultra-log-concavity, and proves ordinary log-concavity when one factor is ultra-log-concave and the other is log-concave with no internal zeros.
The same preprint claims a counterexample to a Fischer–Kubitzke conjecture on Hadamard powers, using Cartesian powers of a Reeve tetrahedron.

September 10, 2026 claimed proof

Yanxin Liu and Jianxi Mao are reported to have proved the conjectured preservation theorem and extended it to finite products and Cartesian products of lattice polytopes. The result is presented as unrefereed and has not been independently verified in the retrieved sources.

Current status (as of September 2026): The conjecture is claimed proved under its stated hypotheses, with extensions, but the proof remains unverified; this does not establish preservation for arbitrary log-concave sequences.

Sources

No solutions have been posted yet.