Eneström-Kakeya Type Theorems in the Complex Numbers and Quaternions.
Abstract
In general, there is no algebraic formula which gives the exact roots of a polynomial of degree greater than four. In light of this, several root-finding algorithms have been developed using tools of numerical analysis, one of the earliest of these being Newton's method. These algorithms can find or approximate individual roots of polynomials. An alternative method is to instead find a region either on the real number line or, more generally, in the complex plane in which all roots must lie. The Eneström-Kakeya Theorem produced an upper bound on the magnitude of the roots of a polynomial whose coefficients satisfy certain conditions. The Eneström-Kakeya Theorem has since been generalized, and various other theorems have been proven which provide upper and lower bounds on the magnitudes of the roots of various kinds of polynomials. Our goal was to generalize these existing theorems further, so that they apply to a broader class of polynomials, both by relaxing the conditions on the coefficients of the polynomials and extending them from the field of real and complex numbers to the quaternionic number system. We employed standard methods of mathematical proof to that end, as well as recent developments in the theory of analytic functions of a quaternionic variable. We proved three new theorems that generalize existing results. Our results are significant in that they continue to push forward a relatively novel area of research.
Start Time
15-4-2026 1:30 PM
End Time
15-4-2026 2:30 PM
Room Number
219
Presentation Type
Oral Presentation
Presentation Subtype
UG Orals
Presentation Category
Science, Technology, and Engineering
Student Type
Undergraduate
Faculty Mentor
Robert Gardner
Eneström-Kakeya Type Theorems in the Complex Numbers and Quaternions.
219
In general, there is no algebraic formula which gives the exact roots of a polynomial of degree greater than four. In light of this, several root-finding algorithms have been developed using tools of numerical analysis, one of the earliest of these being Newton's method. These algorithms can find or approximate individual roots of polynomials. An alternative method is to instead find a region either on the real number line or, more generally, in the complex plane in which all roots must lie. The Eneström-Kakeya Theorem produced an upper bound on the magnitude of the roots of a polynomial whose coefficients satisfy certain conditions. The Eneström-Kakeya Theorem has since been generalized, and various other theorems have been proven which provide upper and lower bounds on the magnitudes of the roots of various kinds of polynomials. Our goal was to generalize these existing theorems further, so that they apply to a broader class of polynomials, both by relaxing the conditions on the coefficients of the polynomials and extending them from the field of real and complex numbers to the quaternionic number system. We employed standard methods of mathematical proof to that end, as well as recent developments in the theory of analytic functions of a quaternionic variable. We proved three new theorems that generalize existing results. Our results are significant in that they continue to push forward a relatively novel area of research.
