On location of zeros of a quaternionic polynomial with restricted coefficients using matrix method

Khangembam Babina Devi, Barchand Chanam

Abstract


In this paper, we consider the connection between zeros of a quaternionic polynomial and left eigen values of its corresponding companion matrix. As an application of which, we determine the region containing the zeros of some special quaternionic polynomials with restricted coefficients. Our result generalizes some previously proved results in this direction. Furthermore, we provide numerical examples and graphical representation to demonstrate the superior precision of our result and its corollary to some previously established results.

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References


A. Aziz, B. A. Zagar, Bounds for the zeros of a polynomial with restricted coefficients, Appl. Math. (Irvine) 3 (2012), 30-33.

N. Carney, R. Gardner, R. Keaton, A. Powers, The Eneström-Kakeya theorem for polynomials of a quaternionic variable, J. Approx. Theory 250 (2020), 1-10.

I. Dar, N. A. Rather, I. Faiq, Bounds on the zeros of a quaternionic polynomial using matrix methods, Filomat 9 (2024), 3001-3010.

S. G. Gal, I. Sabadini, On the Bernstein and Erdős-Lax inequality for quaternionic polynomials, C. R. Acad. Sci. Paris. Ser. I 353 (2015), 5-9.

G. Gentili, C. Stoppato, Zeros of regular functions and polynomials of a quaternionic variable, Michigan Math. J. 56 (2008), 655-667.

G. Gentili, D. C. Struppa, A new theory of regular functions of a quaternionic variable, Adv. Math. 216 (2007), 279-301.

G. Gentili, D. C. Struppa, On the multiplicity of zeros of polynomials with quaternionic coefficients, Milan J. Math. 76 (2008), 15-25.

M. Marden, Geometry of polynomials, Math. Surveys., American Mathematical Society, Providence, RI, 1966.

G. V. Milovanović, A. Mir, Zeros of one class of quaternionic polynomials, Filomat 36 (2022), 6655-6667.

G. V. Milovanović, A. Mir, A. Ahmad, On the zeros of a quaternionic polynomial with restricted coefficients, Linear Algebra Appl. 653 (2022), 231-245.

I. Niven, Equations in quaternions, Am. Math. Mon. 48 (1941), 654-661.

I. Niven, The roots of a quaternion, Am. Math. Mon. 49 (1942), 386-388.

E. R. Nwaeze, Some generalizations of the Eneström-Kakeya theorem, Acta Comment. Univ. Tartu. Math. 20 (2016), 15-21.

N. A. Rather, T. Bhat, I. Dar, B. Khan, A. Sama, Computation of the zeros of a quaternionic polynomial using matrix methods, Arab. J. Basic Appl. Sci. 31 (2024), 65-70.

D. Tripathi, A note on Eneström-Kakeya theorem for a polynomial with quaternionic variable, Arab. J. Math. 9 (2020), 707-714.

B. A. Zargar, M. H. Gulzar, M. Ali, On the zeros of polynomials with restricted coefficients, Ann. Math. Sil. 37 (2023), no. 2, 306-314. DOI:10.2478/amsil-2023-0016




DOI: https://doi.org/10.52846/ami.v52i2.1941