标题： 布兰登关于博罗斯-莫尔多项式的猜想 显示英文标题

标题： Branden's Conjectures on the Boros-Moll Polynomials

摘要： 我们证明了Brändén关于与Boros-Moll多项式$P_n(x)$相关的多项式$Q_n(x)$和$R_n(x)$的实根性的两个猜想。事实上，我们证明了$Q_n(x)$和$R_n(x)$都形成Sturm序列。第一个猜想意味着$P_n(x)$的2-对数凹性，第二个猜想意味着$P_n(x)$的3-对数凹性。 显示英文摘要

摘要： 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 $P_n(x)$.