On Critical Values of Polynomials with Real Critical Points

Abstract

Abstract. Let f be a polynomial of degree at least 2 with f (0) = 0 and f (0) = 1. Suppose that all the zeros of f are real. We show that there is a zero ζ of f such that |f (ζ)/ζ| ≤ 2/3, and that this inequality can be taken to be strict unless f is of the form f (z) = z + cz 3 .

Smale’s Problem for Critical Points on Certain Two Rays

“…Suppose that the critical points of f are contained in the union of k rays from the origin to infinity. In [8] we conjectured that for such a function f , we have S( f ) ≤ 1 − 1/(k + 1), which would imply that K n = 1 − 1/n, and proved that this is true for k = 1 and k = 2.…”

“…In this paper we only consider the case where k = 1 or k = 2. If α = 0 after passing to g, our problem has already been solved in [8]. Therefore we limit ourselves to the case where α = 0.…”

“…If this line passes through the origin, then again we are reduced to the case covered in [8]. Therefore we will assume that under these circumstances this line does not go through the origin.…”

“…Suppose that k = 1, so that there is only one ray which is part of a certain straight line, or that k = 2 and the two rays form a straight line. If this line passes through the origin, then again we are reduced to the case covered in [8]. Therefore we will assume that under these circumstances this line does not go through the origin.…”

“…The idea of the proof of Theorem 1.3 is to reduce the situation to that considered in [8]. There we obtained the following result (see [8, (4)]).…”