International Journal of Mathematics and Mathematical Sciences
Volume 32 (2002), Issue 12, Pages 721-738

Powersum formula for polynomials whose distinct roots are differentially independent over constants

John Michael Nahay

25 Chestnut Hill Lane, Columbus, NJ 08022-1039, USA

Received 20 February 2002

Copyright © 2002 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.


We prove that the author's powersum formula yields a nonzero expression for a particular linear ordinary differential equation, called a resolvent, associated with a univariate polynomial whose coefficients lie in a differential field of characteristic zero provided the distinct roots of the polynomial are differentially independent over constants. By definition, the terms of a resolvent lie in the differential field generated by the coefficients of the polynomial, and each of the roots of the polynomial are solutions of the resolvent. One example shows how the powersum formula works. Another example shows how the proof that the formula is not zero works.