On the Roots of a Family of Polynomials
Abstract
:1. Introduction
- We introduce the family of polynomials obtained by composing the elementary polynomials and .
- We prove that the set of roots of this infinite family of polynomials is bounded by the circles and . A direct consequence is that the roots of the reliability polynomials of series-parallel composition networks are bounded.
- Starting from the result proved by Brown and deGagné [21,23], we show that the closure of the set of roots of the polynomials in contains the domain bounded by the lemniscates with n poles centered at 0 and at 1; hence, it contains two disks of radius slightly greater than 1, centered at 0 and at 1.
- We find 16 complex limit points of the set of roots apart from and , the two limit points on the real axis noted in [21].
2. Preliminaries
- (i)
- (the sets and are symmetric to each other with respect to );
- (ii)
- .
- (iii)
- and , .
- (i)
- If k is even then , and f is strictly increasing on .
- (ii)
- If k is odd then , and f is strictly decreasing on .
3. Bounds for the Set of Roots
4. The Closure of the Set of Roots
5. Limit Points
6. Discussion and Conclusions
- What is the maximum radius of the two disks contained into the closure of the set of roots?
- How can one find other limit points, apart from the ones presented in the last section of the paper?
- The real roots are proved to be in , and the points and are proved to be limit points. Is the set of real roots dense in ?
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
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Jianu, M. On the Roots of a Family of Polynomials. Fractal Fract. 2023, 7, 339. https://doi.org/10.3390/fractalfract7040339
Jianu M. On the Roots of a Family of Polynomials. Fractal and Fractional. 2023; 7(4):339. https://doi.org/10.3390/fractalfract7040339
Chicago/Turabian StyleJianu, Marilena. 2023. "On the Roots of a Family of Polynomials" Fractal and Fractional 7, no. 4: 339. https://doi.org/10.3390/fractalfract7040339
APA StyleJianu, M. (2023). On the Roots of a Family of Polynomials. Fractal and Fractional, 7(4), 339. https://doi.org/10.3390/fractalfract7040339