Moses, Nathan C Approximate Roots and the Hidden Geometry of Polynomial Coefficients <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> approximate roots;polynomial roots;polynomial zeros;polynomial coefficients;coefficient effects;elementary symmetric polynomials;Rouche;rouche's theorem;abhyankar;moh;abhyankar-moh;Complex Polynomials;Algebraic and Differential Geometry;Geometry 2020-03-03
    https://hammer.purdue.edu/articles/thesis/Approximate_Roots_and_the_Hidden_Geometry_of_Polynomial_Coefficients/11874762
10.25394/PGS.11874762.v1