International Journal of Mathematics and Mathematical Sciences
Volume 32 (2002), Issue 12, Pages 721-738
doi:10.1155/S0161171202202331
Powersum formula for polynomials whose distinct roots are differentially independent
over constants
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.
Abstract
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.