J Syst Sci Syst Eng (Sep 2012) 21(3): 372-382 ISSN: 1004-3756 (Paper) 1861-9576 (Online) DOI: 10.1007/s11518-012-5197-y CN11-2983/N  Systems Engineering Society of China and Springer-Verlag Berlin Heidelberg 2012 Technical Note ON DIFFERENTIAL STRUCTURES OF POLYNOMIAL SPACES IN CONTROL THEORY  Baltazar Aguirre HERNÁNDEZ 1 Martn Eduardo FRÍAS-ARMENTA 2,a Fernando VERDUZCO 2,b 1 Depto. de Matemáticas, Universidad Autónoma Metropolitana–Iztapalapa. Av. Sn. Rafael Atlixco no. 186. Col. Vicentina, Iztapalapa 09340, México, D.F., bahe@xanum.uam.mx () 2 Depto. de Matemáticas, Universidad de Sonora. Blvd. Luis Encinas y Rosales s/n. Col. Centro, Hermosillo, Sonora, México. a eduardo@gauss.mat.uson.mx, martineduardofrias@gmail.com b verduzco@gauss.mat.uson.mx 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. Keywords: Schur polynomials, Hurwitz complex polynomials, trivial vector bundle  This work is partly supported by CONACYT CB-2010/150532. 1. Introduction For continuous linear systems, if the characteristic polynomial has all their roots with negative real part, then the origin is stable. By the other hand, for discrete linear systems, if all the roots have magnitude less than one, then the origin is stable. For the continuous case, the characteristic polynomial is called Hurwitz polynomial, while for the discrete case, is called Schur polynomial. The importance of Hurwitz and Schur polynomials in the study of the stability of linear systems has implied that a great amount of information has been produced about them and their properties. The Routh-Hurwitz criterion (Hurwitz 1895) and the Jury’s Test (Jury 1958) are the most used methods for checking the stability in continuous