MORE BOUNDS ON THE LOCATION OF CRITICAL POINTS OF APOLYNOMIAL WITH ALL REAL ZEROS

摘要

1. Introduction. The derivative of a polynomial with all real zeros has a unique simple zero on every interval between two distinct adjacent (not necessarily simple) zeros of that polynomial. However, these critical points cannot lie just anywhere on those intervals. Imagine a situation where two successive distinct zeros of such a polynomial arc fixed and that you have complete freedom of choice for all the other zeros. Different choices for these zeros will of course produce different locations for the zero of the derivative, or critical point, of the polynomial on the aforementioned interval. But can any desired location of the critical point be achieved by appropriately varying the other zeros of the polynomial? The answer turns out to be negative, as is shown by the following result of Peyser ([3]), which was later rediscovered by Andrews ([1]) and which establishes lower and upper bounds on the location of the critical points.