论文标题
关于特殊三项元素零比率的位置
On the location of ratios of zeros of special trinomials
论文作者
论文摘要
给定的codrime整数$ k，\ ell $带有$ k> \ ell \ geqslant 1 $和任意的复杂多项式$ a（z），b（z），b（z）$（a（z）b（z）b（z））\ geqslant \ geqslant 1 $，我们考虑polynomial sequence $ \ \ \ \ {p_n（z）p_n（z） $ p_n（z）+b（z）p_ {n- \ ell}（z）+a（z）p_ {n-k}（z）= 0 $均受初始条件$ p_0（z）= 1 $，$ p _ { - 1}（-1}（z）= \ cdots = \ cdots = p _ p_ {1-k}（z）$ cul cul cul cul $ \ {p_n（z）\} $中的多项式的零。此外，我们表明，对于任何（随机选择的）$ n \ in \ mathbb {z} _ {\ geqslant 1} $和零$ z_0 $ $ p_n（z）$，带有$ a（z_0） $ d（t; z_0）：= {a（z_0）t^{k}+ b（z_0）t^{\ ell} +1} $具有在实际线上的比率，并且在以原点为中心的单位圆圈上。这揭示了由$ t^k+at^{\ ell}+1 $的三位一体的零元素展示的先前未知的几何特性，其中$ a \ in \ mathbb {c} - \ {0 \} $是如此，就是$ a^k \ in \ mathbb {r} $。
Given coprime integers $k, \ell$ with $k > \ell \geqslant 1$ and arbitrary complex polynomials $A(z), B(z)$ with $°(A(z)B(z))\geqslant 1$, we consider the polynomial sequence $\{P_n(z)\}$ satisfying a three-term recurrence $P_n(z)+B(z)P_{n-\ell}(z)+A(z)P_{n-k}(z)=0$ subject to the initial conditions $P_0(z)=1$, $P_{-1}(z)=\cdots=P_{1-k}(z)=0$ and fully characterize the real algebraic curve $Γ$ on which the zeros of the polynomials in $\{P_n(z)\}$ lie. In addition, we show that, for any (randomly chosen) $n\in \mathbb{Z}_{\geqslant 1}$ and zero $z_0$ of $P_n(z)$ with $A(z_0)\neq 0$, at-least two of the distinct zeros of the trinomial $D(t;z_0):={A(z_0)t^{k}+ B(z_0)t^{\ell}+1} $ have a ratio that lies on the real line and / or on the unit circle centred at the origin. This reveals a previously unknown geometric property exhibited by the zeros of trinomials of the form $t^k+at^{\ell}+1$ where $a\in \mathbb{C}-\{0\}$ is such that $a^k\in \mathbb{R}$.
