A、B、D型欧拉多项式之间的半交错性质
The half interlacing property among the types A, B and D Eulerian polynomials
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中文总结 AI 辅助
本文补充了作者2012年关于D型欧拉多项式实根性首个证明的细节，得到A、B、D型欧拉多项式间的半交错性质，完善了该组合多项式领域的相关结论。
中文摘要 AI 辅助
组合多项式理论中有一个著名结果：D型欧拉多项式$D_n(x)$具有实根性，这一性质最初由Brenti于1994年提出猜想。2013年，Savage和Visontai通过在s-逆序列上构造一组相容多项式证明了该猜想；Bränden利用保持非负多项式序列交错性质的矩阵，也证明了$D_n(x)$的实根性；Yang和Zhang结合Hermite-Biehler定理以及Borcea与Brändén关于Hurwitz稳定性的结果，给出了$D_n(x)$实根性的另一证明；Hyatt通过构造D型半欧拉多项式，再次证明了Brenti的猜想。正如Brenti在1994年最初提出的，有可能通过更精确地了解A型和B型欧拉多项式的零点位置，来证明$D_n(x)$的实根性。在本文中，作者对其2012年给出的$D_n(x)$实根性的首个证明补充了更多细节，从而得到了A、B、D型欧拉多项式之间的半交错性质。
英文摘要
A famous result in the theory of combinatorial polynomials is the real-rootedness of the type $D$ Eulerian polynomial $D_n(x)$, which was originally conjectured by Brenti in 1994. By constructing a set of compatible polynomials over $s$-inversion sequences, Savage and Visontai proved this conjecture in 2013. Using matrices preserving interlacing properties of nonnegative polynomial sequences, Bränden also established the real-rootedness of $D_n(x)$. Combining Hermite-Biehler theorem and a result of Borcea and Brändén on Hurwitz stability, Yang and Zhang gave another proof of the real-rootedness of $D_n(x)$. By constructing half Eulerian polynomials of type $D$, Hyatt reproved Brenti's conjecture. As originally suggested by Brenti in 1994, it is possible that the real-rootedness of $D_n(x)$ may be established by using a more precise knowledge of the location of zeros of the types $A$ and $B$ Eulerian polynomials. In this paper, we add more details to the first proof of the real-rootedness of $D_n(x)$ that was provided by the author in 2012, which yields the half interlacing property among the types $A,B$ and $D$ Eulerian polynomials.
