Siebeck curves and two refinements of the

Abstract

We prove two results which improve the well known Gauss-Lucas theorem by locating the roots of the derivative of a complex polynomial f in sets smaller than the convex hull of the roots of f .

Critical points of polynomials

“…Formula (4) with k = 1 shows (since we may assume without loss of generality that the direction w is vertical) that all zeros of P ′ lie in S w , which proves the following theorem of B. Z. Linfield [11] (see also [2]).…”

“…all zeros of P lie in K. Convexity cannot be dropped: if K is closed and not convex, then there are w 1 , w 2 ∈ K such that the line segment connecting w 1 and w 2 lies entirely outside K (except for its endpoints), and then for the polynomial P (z) = (z − w 1 )(z − w 2 ), which has zeros in K, its only critical point (which is the midpoint of segment w 1 w 2 ) lies outside K. Note, however, that for the case K = [−2, −1] ∪ [1,2] all but (possibly) one critical points of a polynomial lie in K if all of its zeros are from K.…”

“…There is a higher-order version (allowing any number of zeros and higher order algebraic curves) due to B. Z. Linfeld [11] (see also [2]).…”

On the critical points of entire functions

“…As for critical points of polynomials (or, more generally, zeros of derivatives of polynomials), the question of determining the distribution and the number of roots of (either real or complex) entire functions has led to many significant results. We refer to [8][9][10][11][12][13] for more on these topics.…”

Roots of complex polynomials and foci of real algebraic curves

“…For the convenience of the reader, the easier case of non-aligned roots is presented first, in Section 6, while the somewhat more technical general case is dealt with in Section 7. Further properties of the Siebeck curve and its application to the location of the roots of the derivative, refining the Gauss-Lucas theorem, will appear in [2].…”