论文标题
特殊的5-期复发关系，带状的toeplitz矩阵和零的现实
Special 5-term recurrence relations, Banded Toeplitz matrices, and Reality of Zeros
论文作者
论文摘要
下面我们建立了保证多项式$ p_n（z）$在多项式序列中的现实的条件 $$ p_ {n}（z）= zp_ {n-1}（z）+αp_{n-2}（z）+βp_{n-3}（z）+γp_{n-4}（n-4}（z），$$，带有标准初始条件$ p_0（z）= 1，p_）= 1，p_）= 1，z） = p _ { - 3}（z）= 0，$ $ $α，β，γ$是真实系数，$γ\ neq 0 $和$ z $是一个复杂的变量。我们将这一一系列多项式解释为适当的带状teoplitz矩阵的主要未成年人，其相关的laurent多项式$ b（z）$是$ \ mathbb {c} \ setMinus \ \ {0 \} $中的holomorphic。我们表明，当$ b（z）$的关键点都是真实的时；或者，当它们是两个真实的和一对复杂的共轭临界点，并在参数上有一些额外的条件时，集合$ b^{ - 1}（\ Mathbb {r}）$包含一个内部装有$ 0 $的Jordan曲线，在某些情况下，非固定曲线包含$ 0 $。对于序列$ \ {p_n（z）\} _ {n = 1}^{\ infty} $中的每个多项式，上述曲线的存在是必要且足够的。
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).
