特殊的5项递归关系，带状Toeplitz矩阵和零点的实性
Special 5-term recurrence relations, Banded Toeplitz matrices, and Reality of Zeros
摘要：
下面我们建立保证多项式$P_n(z)$的所有零点在多项式序列$\{P_n(z)\}_{n=1}^{\infty}$中的真实性的条件，该多项式序列$\{P_n(z)\}_{n=1}^{\infty}$满足具有标准初始条件$$P_0(z) = 1, P_{-1}(z) = P_{-2}(z) =P_{-3}(z) = 0,$$的五项递归关系$$P_{n}(z)= zP_{n-1}(z) + αP_{n-2}(z)+βP_{n-3}(z)+γP_{n-4}(z),$$，其中$α, β, γ$是实系数，$γ\neq 0$和$z$是复变量。我们把这个多项式序列作为适当的带状Teoplitz矩阵的主要未成年人，其相关的Laurent多项式$b(z)$是全纯的$\mathbb{C}\setminus \{0\}$。我们证明了当$b(z)$的临界点都是实的，或者当它们是两个实的和一对复共轭的临界点，并且对参数有一些额外的条件时，集合$b^{-1}(\mathbb{R})$包含一条内部有$0$的Jordan曲线，并且在某些情况下包含一条包围$0$的非简单曲线.所述曲线的存在对于序列$\{P_n(z)\}_{n=1}^{\infty}$中的每个多项式是双曲的（实根的）是必要的和充分的。
英文摘要：
Below we establish the conditions guaranteeing the reality of all the zeros of polynomials $P_n(z)$ in the polynomial sequence $\{P_n(z)\}_{n=1}^{\infty}$ satisfying a five-term recurrence relation $$P_{n}(z)= zP_{n-1}(z) + αP_{n-2}(z)+βP_{n-3}(z)+γP_{n-4}(z),$$ with the standard initial conditions $$P_0(z) = 1, P_{-1}(z) = P_{-2}(z) =P_{-3}(z) = 0,$$ where $α, β, γ$ are real coefficients, $γ\neq 0$ and $z$ is a complex variable. We interprete this sequence of polynomials as principal minors of an appropriate banded Teoplitz matrix whose associated Laurent polynomial $b(z)$ is holomorphic in $\mathbb{C}\setminus \{0\}$. We show that when either the critical points of $b(z)$ are all real; or when they are two real and one pair of complex conjugate critical points with some extra conditions on the parameters, the set $b^{-1}(\mathbb{R})$ contains a Jordan curve with $0$ in its interior and in some cases a non-simple curve enclosing $0$. The presence of the said curves is necessary and sufficient for every polynomial in the sequence $\{P_n(z)\}_{n=1}^{\infty}$ to be hyperbolic (real-rooted).
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