Branden's Conjectures on the Boros-Moll Polynomials

TL;DR

This paper proves two conjectures regarding the real-rootedness and log-concavity properties of polynomials related to Boros-Moll polynomials, establishing Sturm sequence structures and higher-order log-concavity.

Contribution

It confirms Bränden's conjectures on the real-rootedness of polynomials associated with Boros-Moll polynomials and demonstrates their Sturm sequence properties.

Findings

Both $Q_n(x)$ and $R_n(x)$ form Sturm sequences.

The conjectures imply 2-log-concavity and 3-log-concavity of $P_n(x)$.

Proof of real-rootedness for the related polynomials.

Abstract

We prove two conjectures of Br\"{a}nd\'{e}n on the real-rootedness of polynomials $Q_n(x)$ and $R_n(x)$ which are related to the Boros-Moll polynomials $P_n(x)$. In fact, we show that both $Q_n(x)$ and $R_n(x)$ form Sturm sequences. The first conjecture implies the 2-log-concavity of $P_n(x)$, and the second conjecture implies the 3-log-concavity of $P_n(x)$.