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A relative Grace Theorem for complex polynomials

Published online by Cambridge University Press:  11 February 2016

DANIEL PLAUMANN  and
MIHAI PUTINAR
DANIEL PLAUMANN
Affiliation:
Zukunftskolleg and Fachbereich Mathematik, Box 216, Universität Konstanz, 78457 Konstanz, Germany e-mail: Daniel.Plaumann@uni-konstanz.de
MIHAI PUTINAR
Affiliation:
Department of Mathematics, University of California at Santa Barbara, Santa Barbara, CA 93106-3080, U.S.A. School of Mathematics & Statistics, Newcastle University, Newcastle upon Tyne, NE1 7RU, U.K. e-mail: mputinar@math.ucsb.edu, mihai.putinar@ncl.ac.uk
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Abstract

We study the pullback of the apolarity invariant of complex polynomials in one variable under a polynomial map on the complex plane. As a consequence, we obtain variations of the classical results of Grace and Walsh in which the unit disk, or a circular domain, is replaced by its image under the given polynomial map.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 161 , Issue 1 , July 2016 , pp. 17 - 30
Copyright
Copyright © Cambridge Philosophical Society 2016 

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References

REFERENCES

[1]Bernstein, S.Leçons sur les propriétés extrémales et la meilleure approximation des fonctions analytiques d'une variable réelle, X + 208 p. Paris, Gauthier-Villars (Collection de monographies sur la théorie des fonctions), 1926.Google Scholar
[2]Carathéodory, C.Über die gegenseitige Beziehung der Ränder bei der konformen Abbildung des Inneren einer Jordanschen Kurve auf einen Kreis. Math. Ann. 73 (1913), no. 2, 305–320.Google Scholar
[3]Conway, J. B.Functions of one complex variable, II. Graduate Texts in Mathematics, 159 (Springer-Verlag, New York, 1995).Google Scholar
[4]Van der Corput, J. G. and Schaake, G.Ungleichungen für Polynome und trigonometrische Polynome. Compositio Math., 2 (1935), 321–361 German.Google Scholar
[5]Dieudonné, J.La théorie analytique des polynômes d'une variable (a coefficients quelconques). Mem. Sci. Math. Fasc. 93 (Paris: Gauthier-Villars, 1938), 1–71.Google Scholar
[6]Fischer, E.Über die Differentiationsprozesse der Algebra. J. Reine Angew. Math. 148 (1917), 1–78.Google Scholar
[7]Garcia, S. R., Prodan, E. and Putinar, M.Mathematical and physical aspects of complex symmetric operators. J. Phys. A: Math. Gen. 47 (2014).Google Scholar
[8]Grace, J. H. and Young, A.The Algebra of Invariants (Cambridge University Press, Cambridge, 1903).Google Scholar
[9]Hörmander, L.On a theorem of Grace. Math. Scand. 2 (1954), 55–64.Google Scholar
[10]Kreĭn, M. G. and Naĭmark, M. A.The method of symmetric and Hermitian forms in the theory of the separation of the roots of algebraic equations. Linear and Multilinear Algebra 10 (1981), no. 4, 265–308. Translated from the Russian by Boshko, O. and Howland, J. L.Google Scholar
[11]Marden, M.Geometry of polynomials, Second edition. Math. Surv., No. 3. (American Mathematical Society, Providence, R.I., 1966).Google Scholar
[12]Rahman, Q. I. and Schmeisser, G.Analytic theory of polynomials. London Math. Soc. Monogr. New Series, 26 (The Clarendon Press, Oxford University Press, Oxford, 2002).CrossRefGoogle Scholar
[13]Sendov, B. and Sendov, H.Loci of complex polynomials, part I. Trans. Amer. Math. Soc. 366 (2014), no. 10, 5155–5184.Google Scholar
[14]Szegö, G.Bemerkungen zu einem Satz von J. H. Grace über die Wurzeln algebraischer Gleichungen. Math. Z. 13 (1922), no. 1, 28–55 (German).Google Scholar