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
DOI:
https://doi.org/10.1090/S0002-9947-2014-06178-3
Published electronically:
May 30, 2014
MathSciNet review:
3240921
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: The classical Grace theorem states that every circular domain in the complex plane containing the zeros of a polynomial
contains a zero of any of its apolar polynomials. We say that a closed domain
is a locus of
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
and solve it for polynomials of degree
. Numerous examples are given.
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- [2] J.H. Grace, The zeros of a polynomial, Proc. Cambridge Philos. Soc. 11 (1902), 352-357.
- [3] G. Szegö, Bemerkungen zu einem Satz von J. H. Grace über die Wurzeln algebraischer Gleichungen, Math. Z. 13 (1922), no. 1, 28-55 (German). MR 1544526, https://doi.org/10.1007/BF01485280
- [4] David G. Wagner, Multivariate stable polynomials: theory and applications, Bull. Amer. Math. Soc. (N.S.) 48 (2011), no. 1, 53-84. MR 2738906 (2012d:32006), https://doi.org/10.1090/S0273-0979-2010-01321-5
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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:
https://doi.org/10.1090/S0002-9947-2014-06178-3
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.