Differentiation of resultants and common roots of pairs of polynomials

Abstract

The well-known mathematical instrument for detection common roots for pairs of polynomials and multiple roots of polynomials are resultants and discriminants. For a pair of polynomials $f$ and $g$ their resultant $R(f,g)$ is a function of their coefficients. Zeros of resultant $R(f,g)$ correspond to the families of coefficients of $f$ and $g$ such that $f$ and $g$ have a common root. Herewith the calculation of this common root is a separate problem. The principal results on calculation of a unique common root of two polynomials and also about calculating a unique root of multiplicity 2 of a polynomial in terms of the first order partial derivatives of resultants and discriminants are given in the monograph by I.M. Gelfand, M.M. Kapranov, A.V. Zelevinsky [1, Ch. 3, Ch. 12]. A significant development of the ideas of this book in the direction of searching for formulas for multiple roots of polynomials is presented in the paper by I.A. Antipova, E.N. Mikhalkin, A.K. Tsikh [2]. The key result of this article is [2, Theorem 1] where the expression for a unique root of multiplicity $s \geq 3$ in terms of the first order partial derivatives of resultant of the polynomial and it's derivative of order $s-1$. In the present article the explicit formulas for higher derivatives of resultants of pairs of polynomials possessing common roots are obtained. On this basis a series of results that differ in ideas from [2, Theorem 1] linking higher derivatives of resultants and common multiple roots are proven. In addition the results obtained are applied for a new transparent proof of a refinement of [2, Theorem 1].