John Michael Nahay

Copyright © 2004 John Michael Nahay. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

For any univariate polynomial P whose coefficients lie in an ordinary differential field 𝔽 of characteristic zero, and for any constant indeterminate α, there exists a nonunique nonzero linear ordinary differential operator ℜ of finite order such that the αth power of each root of P is a solution of ℜzα=0, and the coefficient functions of ℜ all lie in the differential ring generated by the coefficients of P and the integers ℤ. We call ℜ an α-resolvent of P. The author's powersum formula yields one particular α-resolvent. However, this formula yields extremely large polynomials in the coefficients of P and their derivatives. We will use the A-hypergeometric linear partial differential equations of Mayr and Gelfand to find a particular factor of some terms of this α-resolvent. We will then demonstrate this factorization on an α-resolvent for quadratic and cubic polynomials.