On the Global Continuity of the Roots of Families of Monic Polynomials (in Russian)

TL;DR

This paper investigates the existence of continuous root functions for families of monic polynomials, proving non-existence for certain degrees and coefficient types, and characterizing roots for others.

Contribution

It establishes new results on the global continuity of roots for specific families of monic polynomials with real or complex coefficients.

Findings

No continuous roots for monic quadratic with complex coefficients

No continuous roots for monic quartic and quintic with real coefficients

Existence and characterization of continuous roots for monic quadratic with real coefficients

Abstract

We raise a question on the existence of continuous roots of families of monic polynomials (by the root of a family of polynomials we mean a function of the coefficients of polynomials of a given family that maps each tuple of coefficients to a root of the polynomial with these coefficients). We prove that the family of monic second-degree polynomials with complex coefficients and the families of monic fourth-degree and fifth-degree polynomials with real coefficients have no continuous root. We also prove that the family of monic second-degree polynomials with real coefficients has continuous roots and we describe the set of all such roots.