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Noncommutative rational functions and their finite-dimensional representations : a thesis submitted in fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics, the University of Auckland, 2017
Volčič, Jurij, 1991-
Noncommutative rational functions form a free skew field, a universal object in the category of skew fields. They emerge in various branches of pure and applied mathematics, such as noncommutative ...algebra, automata theory, control theory, free analysis, free real algebraic geometry and free probability. A noncommutative rational function is given by a formal rational expression involving freely noncommuting variables and arithmetic operations, which can be naturally evaluated at tuples of matrices of all sizes. This dissertation studies such finite-dimensional evaluations, with the goal of determining what information about the structure of the free skew field can be recovered from them. Firstly a matrix coefficient realization theory is developed, which for an arbitrary noncommutative rational function yields an efficiently computable normal form. Such a realization of a noncommutative rational function measures its complexity and describes its natural domain as the complement of a free singularity locus of a linear matrix pencil. The inclusion problem for free loci, and thus for domains of noncommutative rational functions, is next solved in terms of epimorphisms between the coefficient algebras of the corresponding linear matrix pencils. This theorem, called a Singularitätstellensatz for linear matrix pencils, is closely related to the invariant theory of the general linear group. Via realization theory this result yields an algebraic characterization of noncommutative rational functions with a given domain. Moreover, a description of noncommutative rational functions whose domains contain all tuples of hermitian matrices is derived. In particular, it is proven that an everywhere defined noncommutative rational function is a noncommutative polynomial. The understanding of the behavior of noncommutative rational functions under hermitian evaluations is further advanced by the resolution of an noncommutative analog of Hilbert's 17th problem: a noncommutative rational function that is positive semidefinite at every tuple of hermitian matrices is a sum of hermitian squares of noncommutative rational functions. Finally, the construction of the free skew field via matrix evaluations is extended to partially commuting arguments. The obtained multipartite rational functions play a remarkable role in the theory of universal skew fields of fractions and in the difference-differential calculus in free analysis.
Type of material - dissertation
; adult, serious
Publication and manufacture - Auckland : [Jurij Volčič], 2017
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