[Paper Review] On annulus containing all the zeros of a polynomial

N. A. Rather, Suhail Gulzar|arXiv (Cornell University)|Nov 30, 2013

Advanced Mathematical Theories and Applications3 references3 citations

TL;DR

This paper presents a new annular region in the complex plane that contains all zeros of a polynomial involving binomial coefficients and generalized Fibonacci numbers, defined by a three-parameter recurrence. By leveraging properties of these generalized Fibonacci sequences and applying inequalities based on coefficient magnitudes, the authors derive tighter bounds than previous results, generalizing Theorems B and C. The key contribution is a unified annulus formula that subsumes earlier results as special cases.

ABSTRACT

In this paper, we obtain an annulus containing all the zeros of the polynomial involving binomial coefficients and generalized Fibonacci numbers. Our result generalize some of the recently obtained results in this direction.

Motivation & Objective

• To derive a tighter annular region containing all zeros of a complex polynomial involving binomial coefficients and generalized Fibonacci numbers.
• To generalize previous results by Díaz-Barrero (2007) and Bidkham et al. (2010) that used standard and t-Fibonacci sequences.
• To unify and extend existing zero localization theorems into a single framework using three positive real parameters (a, b, c) for generalized Fibonacci sequences.
• To provide explicit, computable inner and outer radii for the annulus based on polynomial coefficients and sequence parameters.

Proposed method

• Define a generalized Fibonacci sequence $ F_n^{(a,b,c)} $ with recursive rules based on parity, parameterized by positive reals $ a, b, c $.
• Introduce a key identity (Lemma 2.1) expressing a weighted sum of $ F_k^{(a,b,c)} $ terms as $ F_{4n}^{(a,b,c)} $, crucial for bounding coefficient ratios.
• Establish the outer radius $ r_2 $ using a reverse triangle inequality argument on $ P(z) $, ensuring $ |P(z)| > 0 $ for $ |z| > r_2 $.
• Derive the inner radius $ r_1 $ by applying the outer radius result to the reciprocal polynomial $ Q(z) = z^n P(1/z) $, transforming the zero location problem.
• Use the function $ \xi(k) = k - 2\lfloor k/2 \rfloor $ to handle parity-dependent coefficient scaling in the bounds.
• Combine these bounds into a final annulus $ \{ z \in \mathbb{C} : r_1 \leq |z| \leq r_2 \} $, with $ r_1 $ and $ r_2 $ explicitly defined in terms of $ a,b,c,u,v,w $.

Experimental results

Research questions

• RQ1 Can the zero location bounds for polynomials be tightened by incorporating generalized Fibonacci sequences with three independent parameters?
• RQ2 How do the bounds from Díaz-Barrero (2007) and Bidkham et al. (2010) relate to a broader class of generalized Fibonacci sequences?
• RQ3 What is the optimal annular region containing all zeros of a polynomial when coefficients are combined with generalized Fibonacci terms and binomial coefficients?
• RQ4 Can a single formula unify previous results based on standard and t-Fibonacci sequences?
• RQ5 What is the quantitative improvement in zero localization bounds when using the generalized framework compared to earlier methods?

Key findings

• The proposed annulus $ \{ z \in \mathbb{C} : r_1 \leq |z| \leq r_2 \} $ contains all zeros of any non-constant complex polynomial of degree $ n $, with $ r_1 $ and $ r_2 $ explicitly defined using generalized Fibonacci numbers $ F_k^{(a,b,c)} $ and binomial coefficients.
• The inner radius $ r_1 $ is given by $ \frac{uv+2w}{uvw+w^2} \min_k \left\{ \frac{(uvw+w^2)^n u^{\xi(k)} (uv)^{\lfloor k/2 \rfloor} F_k^{(u,v,w)} \binom{n}{k}}{F_{4n}^{(u,v,w)}} \left| \frac{a_0}{a_k} \right| \right\}^{1/k} $, where $ \xi(k) = k - 2\lfloor k/2 \rfloor $.
• The outer radius $ r_2 $ is given by $ \frac{abc + c^2}{ab + 2c} \max_k \left\{ \frac{F_{4n}^{(a,b,c)}}{(abc + c^2)^n a^{\xi(k)} (ab)^{\lfloor k/2 \rfloor} F_k^{(a,b,c)} \binom{n}{k}} \left| \frac{a_{n-k}}{a_n} \right| \right\}^{1/k} $.
• The result generalizes Theorem B (Díaz-Barrero, 2007) and Theorem C (Bidkham et al., 2010), which are recovered as special cases by setting $ a = b = u = v = t $, $ c = w = 1 $.
• For the polynomial $ P(z) = z^3 + 0.1z^2 + 0.3z + 0.7 $, the new bound yields $ |z| < 1.185 $, improving upon the previous upper bound of $ 1.23 $ from Theorem B.
• The method ensures $ |P(z)| > 0 $ for $ |z| > r_2 $, proving all zeros lie within the closed disk $ |z| \leq r_2 $, and duality via the reciprocal polynomial ensures all zeros lie outside $ |z| < r_1 $.