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Approximate Roots and the Hidden Geometry of Polynomial Coefficients

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posted on 03.03.2020 by Nathan C Moses
The algebraic operation of approximate roots provides a geometric approximation of the zeros of a polynomial in the complex plane given conditions on their symmetry. A polynomial of degree n corresponds to a cluster of n zeros in the complex plane. The zero of the nth approximate root polynomial locates the gravitational center of this cluster. When the polynomial is of degree mn, with m clusters of n zeros, the centers of the clusters are no longer identified by the zeros of the nth approximate root polynomial in general. The approximation of the centers can be recovered given assumptions about the symmetric distribution of the zeros within each cluster, and given that m > n. Rouch´e’s theorem is used to extend this result to relax some of these conditions. This suggests an insight into the geometry of the distribution of zeros within the complex plane hidden within the coefficients of polynomials.

History

Degree Type

Doctor of Philosophy

Department

Mathematics

Campus location

West Lafayette

Advisor/Supervisor/Committee Chair

Dr. Tzuong-Tsieng Moh

Advisor/Supervisor/Committee co-chair

Dr. William Heinzer

Additional Committee Member 2

Dr. Tong Liu

Additional Committee Member 3

Dr. Louis de Branges

Licence

Exports