Lin–Ma–Ma–Zhou real-rootedness conjecture

For every admissible multiset parameter $\mathbf{s}$ , the associated multiset Eulerian–Narayana polynomial $E_{\mathbf{s}}(t)$ is real-rooted. More precisely, after removing its factor $t$ , all remaining zeros are simple and negative; equivalently, $E_{\mathbf{s}}(t)/t$ has only simple zeros in $(-\infty,0)$ .

References

Primary source

arXiv

Additional references

Zeros and interlacing for multiset Eulerian-Narayana polynomials — arXiv — Philip B. Zhang, Tongyuan Zhao

algebraic-combinatorics polynomial-theory

Posted 5 days ago · edited 5 days ago

Progress summary

Refreshed 5 days ago

Claimed solved

A new preprint claims to settle the conjecture, but the claim has not been independently checked.

The Lin–Ma–Ma–Zhou conjecture predicts real-rootedness for the relevant multiset Eulerian–Narayana polynomials. The reported preprint claims this and several stronger consequences.

October 1, 2026 preprint

Philip B. Zhang and Tongyuan Zhao claim to prove the conjectured real-rootedness, together with monotonicity, interlacing, and transition-matrix properties. The result is presented as a resolution, but the retrieved record supplies no independent expert assessment.

Current status (as of October 2026): The conjecture is claimed proved by Zhang and Zhao, but the unrefereed preprint has not been independently verified.

Sources

No solutions have been posted yet.