On the location of ratios of zeros of special trinomials
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| Title: | On the location of ratios of zeros of special trinomials |
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| Authors: | Bamunoba, Alex Samuel, Ndikubwayo, Innocent |
| Source: | Quaestiones Mathematicae. :1-24 |
| Publication Status: | Preprint |
| Publisher Information: | National Inquiry Services Center (NISC), 2025. |
| Publication Year: | 2025 |
| Subject Terms: | Mathematics - Complex Variables, FOS: Mathematics, 0101 mathematics, Complex Variables (math.CV), 01 natural sciences, Primary 12D10, Secondary 26C10, 30C15 |
| Description: | 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 $a^k\in \mathbb{R}$. 17 pages, 4 figures, 1 table |
| Document Type: | Article |
| Language: | English |
| ISSN: | 1727-933X 1607-3606 |
| DOI: | 10.2989/16073606.2025.2547044 |
| DOI: | 10.48550/arxiv.2210.06403 |
| Access URL: | http://arxiv.org/abs/2210.06403 |
| Rights: | arXiv Non-Exclusive Distribution |
| Accession Number: | edsair.doi.dedup.....d43ac0f945389791bb2bc0f81b7424ff |
| Database: | OpenAIRE |
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Nájsť tento článok vo Web of Science | Abstract: | 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 $a^k\in \mathbb{R}$.<br />17 pages, 4 figures, 1 table |
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| ISSN: | 1727933X 16073606 |
| DOI: | 10.2989/16073606.2025.2547044 |