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Innocent Ndikubwayo

Publications and source records attributed to Innocent Ndikubwayo.

6 recordsLinked to original sources
arXiv2024

On the location of ratios of zeros of special trinomials

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 $a^k\in \mathbb{R}$.

math.CV↗
arXiv2020

Special 5-term recurrence relations, Banded Toeplitz matrices, and Reality of Zeros

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).

math.CV↗
arXiv2020

Non-real zeros of polynomials in a polynomial sequence satisfying a three-term recurrence relation

This paper discusses the location of zeros of polynomials in a polynomial sequence $\{P_n(z)\}$ generated by a three-term recurrence relation of the form $P_n(z)+ B(z)P_{n-1}(z) +A(z) P_{n-k}(z)=0$ with $k>2$ and the standard initial conditions $P_{0}(z)=1, P_{-1}(z)=\ldots=P_{-k+1}(z)=0,$ where $A(z)$ and $B(z)$ are arbitrary coprime real polynomials. We show that there always exist polynomials in $\{P_n(z)\}$ with non-real zeros.

math.CV↗
arXiv2020

Generalizing Tran's Conjecture

A conjecture of Khang Tran [6] claims that for an arbitrary pair of polynomials $A(z)$ and $B(z)$, every zero of every polynomial in the sequence $\{P_n(z)\}_{n=1}^\infty$ satisfying the three-term recurrence relation of length $k$ $$P_n(z)+B(z)P_{n-1}(z)+A(z)P_{n-k}(z)=0 $$ with the standard initial conditions $P_0(z)=1$, $P_{-1}(z)=\dots=P_{-k+1}(z)=0$ which is not a zero of $A(z)$ lies on the real (semi)-algebraic curve $\mathcal C \subset \mathbb {C}$ given by $$\Im \left(\frac{B^k(z)}{A(z)}\right)=0\quad {\rm and}\quad 0\le (-1)^k\Re \left(\frac{B^k(z)}{A(z)}\right)\le \frac{k^k}{(k-1)^{k-1}}.$$ In this short note, we show that for the recurrence relation (generalizing the latter recurrence of Tran) given by $$P_n(z)+B(z)P_{n-\ell}(z)+A(z)P_{n-k}(z)=0, $$ with coprime $k$ and $\ell$ and the same standard initial conditions as above, every root of $P_n(z)$ which is not a zero of $A(z)B(z)$ belongs to the real algebraic curve $\mathcal C_{\ell,k}$ given by $$\Im \left(\frac{B^k(z)}{A^\ell(z)}\right)=0.$$

math.CA↗
arXiv2019

Around a Conjecture of K. Tran

We study the root distribution of a sequence of polynomials $\{P_n(z)\}_{n=0}^{\infty}$ with the rational generating function $$ \sum_{n=0}^{\infty} P_n(z)t^n= \frac{1}{1+ B(z)t^\ell +A(z)t^k}$$ for $(k,\ell)=(3,2)$ and $(4,3)$ where $A(z)$ and $B(z)$ are arbitrary polynomials in $z$ with complex coefficients. We show that the zeros of $P_n(z)$ which satisfy $A(z)B(z)\neq 0$ lie on a real algebraic curve which we describe explicitly.

math.CV↗