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A converse to the Grace–Walsh–Szegő theorem

Published online by Cambridge University Press:  01 September 2009

PETTER BRÄNDÉN  and
DAVID G. WAGNER
PETTER BRÄNDÉN
Affiliation:
Department of Mathematics, Royal Institute of Technology, SE-100 44 Stockholm, Sweden. e-mail: [email protected]
DAVID G. WAGNER
Affiliation:
Department of Combinatorics and Optimization, University of Waterloo, Ontario, CanadaN2L 3G1. e-mail: [email protected]
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Abstract

We prove that the symmetrizer of a permutation group preserves stability if and only if the group is orbit homogeneous. A consequence is that the hypothesis of permutation invariance in the Grace–Walsh–Szegő Coincidence Theorem cannot be relaxed. In the process we obtain a new characterization of the Grace-like polynomials, introduced by D. Ruelle, and prove that the class of such polynomials can be endowed with a natural multiplication.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 147 , Issue 2 , September 2009 , pp. 447 - 453
Copyright
Copyright © Cambridge Philosophical Society 2009

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References

REFERENCES

[1]Beaumont, R. A. and Peterson, R. P.Set-transitive permutation groups. Canad. J. Math. 7 (1955), 35–42.CrossRefGoogle Scholar
[2]Borcea, J. and Brändén, P. The Lee–Yang and Pólya–Schur programs. I. Linear operators preserving stability. arXiv:0809.0401.Google Scholar
[3]Cameron, P. J. and Dent, A. W.Orbit homogeneity in permutation groups. Bull. London Math. Soc. 38 (2006), 587–596.CrossRefGoogle Scholar
[4]Choe, Y., Oxley, J., Sokal, A. and Wagner, D. G.Homogeneous multivariate polynomials with the half-plane property. Adv. Appl. Math. 32 (2004), 88–187.CrossRefGoogle Scholar
[5]Grace, J. H.The zeros of a polynomial. Proc. Camb. Phil. Soc. 11 (1902), 352–357.Google Scholar
[6]Hörmander, L.On a theorem of Grace. Math. Scand. 2 (1954), 55–64.CrossRefGoogle Scholar
[7]Rahman, Q. I. and Schmeisser, G.Analytic Theory of Polynomials. (Clarendon Press, 2002).CrossRefGoogle Scholar
[8]Ruelle, D. Grace-like polynomials. in “Foundations of computational mathematics (Hong Kong 2000)” pp. 405–421. (World Sci. Publ. Co. Inc., 2002).CrossRefGoogle Scholar
[9]Szegő, G.Bemerkungen zu einem Satz von J. H. Grace über die Wurzeln algebraischer Gleichungen. Math. Z. 13 (1922), 28–55.CrossRefGoogle Scholar
[10]Walsh, J. L.On the location of the roots of certain types of polynomials. Trans. Amer. Math. Soc. 24 (1922), 163–180.CrossRefGoogle Scholar