Hilbert’s 17th problem in free skew fields

Forum of Mathematics Sigma ◽

10.1017/fms.2021.54 ◽

2020 ◽

Vol 9 ◽

Author(s):

Jurij Volčič

Keyword(s):

Rational Function ◽

Extension Theorem ◽

Rational Functions ◽

Positive Semidefinite ◽

Linear Matrix ◽

Intermediate Step ◽

Noncommutative Polynomials ◽

Skew Fields ◽

Matrix Pencils ◽

17Th Problem

Abstract This paper solves the rational noncommutative analogue of Hilbert’s 17th problem: if a noncommutative rational function is positive semidefinite on all tuples of Hermitian matrices in its domain, then it is a sum of Hermitian squares of noncommutative rational functions. This result is a generalisation and culmination of earlier positivity certificates for noncommutative polynomials or rational functions without Hermitian singularities. More generally, a rational Positivstellensatz for free spectrahedra is given: a noncommutative rational function is positive semidefinite or undefined at every matricial solution of a linear matrix inequality $L\succeq 0$ if and only if it belongs to the rational quadratic module generated by L. The essential intermediate step toward this Positivstellensatz for functions with singularities is an extension theorem for invertible evaluations of linear matrix pencils.

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Reduced unitary Whitehead groups of skew fields of noncommutative rational functions

Journal of Soviet Mathematics ◽

10.1007/bf01476123 ◽

1982 ◽

Vol 19 (1) ◽

pp. 1067-1071 ◽

Cited By ~ 2

Author(s):

V. I. Yanchevskii

Keyword(s):

Rational Functions ◽

Skew Fields ◽

Whitehead Groups

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Value distribution of meromorphic functions concerning rational functions and differences

Advances in Difference Equations ◽

10.1186/s13662-020-03150-6 ◽

2020 ◽

Vol 2020 (1) ◽

Author(s):

Mingliang Fang ◽

Degui Yang ◽

Dan Liu

Keyword(s):

Positive Integer ◽

Meromorphic Function ◽

Rational Function ◽

Finite Order ◽

Meromorphic Functions ◽

Rational Functions ◽

Value Distribution ◽

Exponent Of Convergence ◽

Nonzero Constant ◽

Zero Points

AbstractLet c be a nonzero constant and n a positive integer, let f be a transcendental meromorphic function of finite order, and let R be a nonconstant rational function. Under some conditions, we study the relationships between the exponent of convergence of zero points of $f-R$ f − R , its shift $f(z+nc)$ f ( z + n c ) and the differences $\Delta _{c}^{n} f$ Δ c n f .

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Rank-one solutions for homogeneous linear matrix equations over the positive semidefinite cone

Applied Mathematics and Computation ◽

10.1016/j.amc.2012.11.022 ◽

2013 ◽

Vol 219 (10) ◽

pp. 5569-5583 ◽

Cited By ~ 1

Author(s):

Yun-Bin Zhao ◽

Masao Fukushima

Keyword(s):

Positive Semidefinite ◽

Linear Matrix ◽

Matrix Equations ◽

Positive Semidefinite Cone ◽

Rank One ◽

Homogeneous Linear ◽

Semidefinite Cone ◽

Linear Matrix Equations

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On the Cauchy index of a real rational function and the index theory of pseudo-lossless rational functions

Stability Theory ◽

10.1007/978-3-0348-9208-7_4 ◽

1996 ◽

pp. 23-31

Author(s):

Yves V. Genin

Keyword(s):

Rational Function ◽

Index Theory ◽

Rational Functions ◽

Cauchy Index

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Free Function Theory Through Matrix Invariants

Canadian Journal of Mathematics ◽

10.4153/cjm-2015-055-7 ◽

2017 ◽

Vol 69 (02) ◽

pp. 408-433 ◽

Cited By ~ 6

Author(s):

Igor Klep ◽

Špela Špenko

Keyword(s):

Power Series ◽

Analytic Functions ◽

Function Theory ◽

Rational Functions ◽

Generalized Polynomials ◽

Direct Sums ◽

Real Analytic Functions ◽

Power Series Expansions ◽

Noncommutative Polynomials ◽

Free Function

Abstract This paper concerns free function theory. Freemaps are free analogs of analytic functions in several complex variables and are defined in terms of freely noncommuting variables. A function of g noncommuting variables is a function on g-tuples of square matrices of all sizes that respects direct sums and simultaneous conjugation. Examples of such maps include noncommutative polynomials, noncommutative rational functions, and convergent noncommutative power series. In sharp contrast to the existing literature in free analysis, this article investigates free maps with involution, free analogs of real analytic functions. To get a grip on these, techniques and tools from invariant theory are developed and applied to free analysis. Here is a sample of the results obtained. A characterization of polynomial free maps via properties of their finite-dimensional slices is presented and then used to establish power series expansions for analytic free maps about scalar and non-scalar points; the latter are series of generalized polynomials for which an invarianttheoretic characterization is given. Furthermore, an inverse and implicit function theorem for free maps with involution is obtained. Finally, with a selection of carefully chosen examples it is shown that free maps with involution do not exhibit strong rigidity properties enjoyed by their involutionfree counterparts.

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The depth of continued fraction expansion for some classes of rational functions

Asian-European Journal of Mathematics ◽

10.1142/s1793557121500339 ◽

2020 ◽

pp. 2150033

Author(s):

Yanapat Tongron ◽

Narakorn Rompurk Kanasri ◽

Vichian Laohakosol

Keyword(s):

Rational Function ◽

Upper Bound ◽

Continued Fraction ◽

Rational Functions ◽

Linear Fractional Transformation ◽

Continued Fraction Expansion ◽

Polynomial Formula ◽

Nonzero Polynomial

For nonzero polynomials [Formula: see text] and [Formula: see text] over a field [Formula: see text], let [Formula: see text] be the depth (length) of the continued fraction expansion for [Formula: see text]. An upper bound on [Formula: see text], for nonzero polynomial [Formula: see text] and rational function [Formula: see text] is obtained. Applying this result, an upper bound on the depth of a linear fractional transformation is also established.

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Any counterexample to Makienko’s conjecture is an indecomposable continuum

Ergodic Theory and Dynamical Systems ◽

10.1017/s014338570800059x ◽

2009 ◽

Vol 29 (3) ◽

pp. 875-883 ◽

Cited By ~ 1

Author(s):

CLINTON P. CURRY ◽

JOHN C. MAYER ◽

JONATHAN MEDDAUGH ◽

JAMES T. ROGERS Jr

Keyword(s):

Rational Function ◽

Julia Set ◽

Julia Sets ◽

Rational Functions ◽

Fatou Set ◽

Indecomposable Continuum

AbstractMakienko’s conjecture, a proposed addition to Sullivan’s dictionary, can be stated as follows: the Julia set of a rational function R:ℂ∞→ℂ∞ has buried points if and only if no component of the Fatou set is completely invariant under the second iterate of R. We prove Makienko’s conjecture for rational functions with Julia sets that are decomposable continua. This is a very broad collection of Julia sets; it is not known if there exists a rational function whose Julia set is an indecomposable continuum.

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Minimal models for rational functions in a dynamical setting

LMS Journal of Computation and Mathematics ◽

10.1112/s1461157012001131 ◽

2012 ◽

Vol 15 ◽

pp. 400-417

Author(s):

Nils Bruin ◽

Alexander Molnar

Keyword(s):

Rational Function ◽

Rational Functions ◽

Minimal Models ◽

Linear Transformations ◽

Single Orbit ◽

Practical Algorithm

AbstractWe present a practical algorithm to compute models of rational functions with minimal resultant under conjugation by fractional linear transformations. We also report on a search for rational functions of degrees 2 and 3 with rational coefficients that have many integers in a single orbit. We find several minimal quadratic rational functions with eight integers in an orbit and several minimal cubic rational functions with ten integers in an orbit. We also make some elementary observations on possibilities of an analogue of Szpiro’s conjecture in a dynamical setting and on the structure of the set of minimal models for a given rational function.

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