On differential structures of polynomial spaces in control theory |
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Authors: | Baltazar Aguirre Hernández Martn Eduardo Frías-Armenta Fernando Verduzco |
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Institution: | Baltazar Aguirre HERNANDEZ1 Martn Eduardo FRIAS-ARMENTA(2,a) Fernando VERDUZCO(2,b) 1 Depto.de Matemdticas.Universidad Autonoma Metropolilana-lztapalapa. Av.Sn.Rafael Atlixco no.186.Col.Vicentina,Iztapalapa 09340,Mexico,D.F., 2 Depto.de Matemdticas,Universidad de Sonora. Blvd.Luis Encinas y Rosales s/n.Col.Centro,Hermosillo,Sonora,Mexico. |
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Abstract: | A valuable number of works has been published about Hurwitz and Schur polynomials in order to known better their properties. For example it is known that the sets of Hurwitz and Schur polynomials are open and no convex sets. Besides, the set of monic Schur polynomials is contractible. Now we study this set using ideas from differential topology, and we prove that the space of Schur complex polynomials with positive leading coefficient, and the space of Hurwitz complex polynomials which leading coefficient having positive real part, have structure of trivial vector bundle, and each space of (Schur complex and real, Hurwitz complex) polynomials has a differential structure diffeomorphic to some known spaces. |
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Keywords: | Schur polynomials Hurwitz complex polynomials trivial vector bundle |
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