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Sector Analogue of the Gauss–Lucas Theorem

Part of: Geometric function theory

Published online by Cambridge University Press:  12 December 2019

Blagovest Sendov  and
Hristo Sendov
Blagovest Sendov
Affiliation:
Bulgarian Academy of Sciences, Institute of Information and Communication Technologies, Acad. G. Bonchev Str., bl. 25A, 1113 Sofia, Bulgaria
Hristo Sendov
Affiliation:
Department of Statistical and Actuarial Sciences, Western University, 1151 Richmond Str., London, ON, N6A 5B7 Canada Email: hssendov@stats.uwo.ca
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Abstract

The classical Gauss–Lucas theorem states that the critical points of a polynomial with complex coefficients are in the convex hull of its zeros. This fundamental theorem follows from the fact that if all the zeros of a polynomial are in a half plane, then the same is true for its critical points. The main result of this work replaces the half plane with a sector as follows.

We show that if the coefficients of a monic polynomial $p(z)$ are in the sector {tei𝜓 : 𝜓∈ [0, 𝜙], t⩾0}, for some $\unicode[STIX]{x1D719}\in [0,\unicode[STIX]{x1D70B})$, and the zeros are not in its interior, then the critical points of $p(z)$ are also not in the interior of that sector.

In addition, we give a necessary condition for a polynomial to satisfy the premise of the main result.

Keywords

Gauss–Lucas theorempolynomialzeros and critical points of polynomialpolynomial with coefficients in a sectorinterlacing polynomialnon-convex

MSC classification

Primary: 30C10: Polynomials
Type
Article
Information
Canadian Journal of Mathematics , Volume 73 , Issue 2 , April 2021 , pp. 318 - 338
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Copyright
© Canadian Mathematical Society 2019

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Footnotes

The first author was partly supported by the Bulgarian National Science Fund under project FNI I 20/20 “Efficient Parallel Algorithms for Large-Scale Computational Problems”. The second author was partially supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada.

†

Deceased.

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