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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Loci of complex polynomials, part I


Authors: Blagovest Sendov and Hristo Sendov
Journal: Trans. Amer. Math. Soc. 366 (2014), 5155-5184
MSC (2010): Primary 30C10
Published electronically: May 30, 2014
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Abstract: The classical Grace theorem states that every circular domain in the complex plane $ \mathbb{C}$ containing the zeros of a polynomial $ p(z)$ contains a zero of any of its apolar polynomials. We say that a closed domain $ \Omega \subseteq \mathbb{C}^*$ is a locus of $ p(z)$ if it contains a zero of any of its apolar polynomials and is the smallest such domain with respect to inclusion. In this work we establish several general properties of the loci and show, in particular, that the property of a set being a locus of a polynomial is preserved under a Möbius transformation. We pose the problem of finding a locus inside the smallest disk containing the roots of $ p(z)$ and solve it for polynomials of degree $ 3$. Numerous examples are given.


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Additional Information

Blagovest Sendov
Affiliation: Bulgarian Academy of Sciences, Institute of Information and Communication Technologie, Acad. G. Bonchev Str., bl. 25A, 1113 Sofia, Bulgaria
Email: acad@sendov.com

Hristo Sendov
Affiliation: Department of Statistical and Actuarial Sciences, Western University, 1151 Richmond Str., London, Ontario, Canada N6A 5B7
Email: hssendov@stats.uwo.ca

DOI: http://dx.doi.org/10.1090/S0002-9947-2014-06178-3
PII: S 0002-9947(2014)06178-3
Keywords: Polynomial, locus, locus of a polynomial, apolar polynomial, multiaffine symmetric polynomial, Grace theorem, Grace-Walsh-Szeg\H{o} coincidence theorem, M\"obius transformation
Received by editor(s): July 9, 2012
Published electronically: May 30, 2014
Additional Notes: The first author was partially supported by Bulgarian National Science Fund #DTK 02/44
The second author was partially supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.