Polynomials in Control Theory Parametrized by Their Roots

descriptionPublicationkeyboard_double_arrow_right Article , Other literature type 01 Jan 2012 Italy English Publisher:Hindawi LimitedJournal:International Journal of Mathematics and Mathematical Sciences, volume 2,012, pages 1-19 (issn: 0161-1712, eissn: 1687-0425,

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Authors: Baltazar Aguirre‐Hernández; José Luis Cisneros‐Molina; Martín-Eduardo Frías-Armenta;

doi: 10.1155/2012/595076 , 10.60692/18n06-02r10 , 10.60692/9q07e-nvx39

Polynomials in Control Theory Parametrized by Their Roots

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Abstract

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.

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Keywords

Contractible space, Space (punctuation), Polynomial, Mathematical analysis, Mathematics (miscellaneous), Ergodic Theory, QA1-939, FOS: Mathematics, Complex quadratic polynomial, Aperiodic graph, Mathematical Physics, Algebra over a field, Applied Mathematics, Pure mathematics, Linguistics, Discrete mathematics, Quaternionic Analysis and Applications, FOS: Philosophy, ethics and religion, Philosophy, Boundary (topology), Combinatorics, Algebraic methods, Physical Sciences, Dynamical Systems and Chaos Theory, FOS: Languages and literature, Monic polynomial, Homeomorphism (graph theory), p-adic Models in Mathematical Physics, Polynomials in general fields (irreducibility, etc.), Mathematics

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