A converse to the Grace--Walsh--Szegő theorem

Details A converse to the Grace--Walsh--Szegő theorem A converse to the Grace--Walsh--Szegő theorem

2008-09-18

arxiv.org/abs/0809.3225v1

Mathematics

Complex Variables

7 pages

Scientific paper

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

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