Roots and polynomials as Homeomorphic spaces
Copyright policy )Roots and polynomials as Homeomorphic spaces
We provide a unified, elementary, topological approach to the classical results stating the continuity of the complex roots of a polynomial with respect to its coefficients, and the continuity of the coefficients with respect to the roots. In fact, endowing the space of monic polynomials of a fixed degree $n$ and the space of $n$ roots with suitable topologies, we are able to formulate the classical theorems in the form of a homeomorphism. Related topological facts are also considered.
16 pages
- Western Washington University United States
- Eastern Washington University United States
- Ball State University United States
continuity of roots wrt the coefficients, Mathematics(all), 30C15, 26C10, General Topology (math.GN), Real polynomials: location of zeros, General Mathematics (math.GM), Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral), FOS: Mathematics, Mathematics - General Mathematics, Roots of polynomials, Homeomorphism, Continuity, Mathematics - General Topology
continuity of roots wrt the coefficients, Mathematics(all), 30C15, 26C10, General Topology (math.GN), Real polynomials: location of zeros, General Mathematics (math.GM), Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral), FOS: Mathematics, Mathematics - General Mathematics, Roots of polynomials, Homeomorphism, Continuity, Mathematics - General Topology
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