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

Research paper by Evgeny E. Bukzhalev

Indexed on: 21 Sep '17Published on: 21 Sep '17Published in: arXiv - Mathematics - Classical Analysis and ODEs



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.