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International Journal of Mathematics and Mathematical Sciences
Volume 2012, Article ID 595076, 19 pages
Research Article

Polynomials in Control Theory Parametrized by Their Roots

1Departamento de Matemáticas, División de Ciencias Básicas e Ingeniería, Universidad Autónoma Metropolitana-Iztapalapa, Avenida San Rafael Atlixco no. 186, Colonia Vicentina, 09340 Mexico, DF, Mexico
2Instituto de Matemáticas, Unidad Cuernavaca, Universidad Nacional Autónoma de México, Avenida Universidad s/n, Colonia Lomas de Chamilpa, 62210 Cuernavaca, MOR, Mexico
3Mathematics Section, The Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, 34014 Trieste, Italy
4Departamento de Matemáticas, División de Ciencias Exactas, Universidad de Sonora, Boulevard Luis Encinas y Rosales s/n, Colonia Centro, 83000 Hermosillo, SON, Mexico

Received 25 May 2012; Accepted 26 August 2012

Academic Editor: Irena Lasiecka

Copyright © 2012 Baltazar Aguirre-Hernández et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The aim of this paper is to introduce the space of roots to study the topological properties of the spaces of polynomials. Instead of identifying a monic complex polynomial with the vector of its coefficients, we identify it with the set of its roots. Viète's map gives a homeomorphism between the space of roots and the space of coefficients and it gives an explicit formula to relate both spaces. Using this viewpoint we establish that the space of monic (Schur or Hurwitz) aperiodic polynomials is contractible. Additionally we obtain a Boundary Theorem.