10.25394/PGS.11874762.v1
Nathan C Moses
Approximate Roots and the Hidden Geometry of Polynomial Coefficients
2020
Purdue University Graduate School
approximate roots
polynomial roots
polynomial zeros
polynomial coefficients
coefficient effects
elementary symmetric polynomials
Rouche
rouche's theorem
abhyankar
moh
abhyankar-moh
Complex Polynomials
2020-03-03 20:06:41
article
https://hammer.figshare.com/articles/thesis/Approximate_Roots_and_the_Hidden_Geometry_of_Polynomial_Coefficients/11874762
<div>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.</div>