Separation Theorems for Group Invariant Polynomials |
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Authors: | Richard M Aron Javier Falcó Manuel Maestre |
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Institution: | 1.Department of Mathematical Sciences,Kent State University,Kent,USA;2.Département de Mathématique,Université de Mons,Mons,Belgium;3.Departamento de Análisis Matemático,Universidad de Valencia,Burjasot, Valencia,Spain |
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Abstract: | We study the existence of separation theorems by polynomials that are invariant under a group action. We show that if G is a finite subgroup of \(\textit{GL}(n,{\mathbb {C}})\), K is a set in \({\mathbb {C}}^{n}\) that is invariant under the action of G and z is a point in \({\mathbb {C}}^{n}\setminus K\) that can be separated from K by a polynomial Q, then z can be separated from K by a G-invariant polynomial P. Furthermore, if Q is homogeneous then P can be chosen to be homogeneous. As a particular case, if K is a symmetric polynomially convex compact set in \({\mathbb {C}}^{n}\) and \(z\notin K\) then there exists a symmetric polynomial that separates z and K. |
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