On the location of ratios of zeros of special trinomials
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866915008222330880 |
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| author | Bamunoba, Alex Samuel Ndikubwayo, Innocent |
| author_facet | Bamunoba, Alex Samuel Ndikubwayo, Innocent |
| contents | 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 $a^k\in \mathbb{R}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2210_06403 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On the location of ratios of zeros of special trinomials Bamunoba, Alex Samuel Ndikubwayo, Innocent Complex Variables Primary 12D10, Secondary 26C10, 30C15 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 $a^k\in \mathbb{R}$. |
| title | On the location of ratios of zeros of special trinomials |
| topic | Complex Variables Primary 12D10, Secondary 26C10, 30C15 |
| url | https://arxiv.org/abs/2210.06403 |