Abstract. In this paper the stable extended domain of a noncommutative rational function is introduced and it is shown that it can be completely described by a monic linear pencil from the minimal realization of the function. This result amends the singularities theorem of Kalyuzhnyi-Verbovetskyi and Vinnikov. Furthermore, for noncommutative rational functions which are regular at a scalar point it is proved that their domains and stable extended domains coincide.

Noncommutative rational functions, i.e., elements of the universal skew field of fractions of a free algebra, can be defined through evaluations of noncommutative rational expressions on tuples of matrices. This interpretation extends their traditionally important role in the theory of division rings and gives rise to their applications in other areas, from free real algebraic geometry to systems and control theory. If a noncommutative rational function is regular at the origin, it can be described by a linear object, called a realization. In this article we present an extension of the realization theory that is applicable to arbitrary noncommutative rationas functions and is welladapted for studying matrix evaluations.Of special interest are the minimal realizations, which compensate the absence of a canonical form for noncommutative rational functions. The non-minimality of a realization is assessed by obstruction modules associated with it; they enable us to devise an efficient method for obtaining minimal realizations. With them we describe the stable extended domain of a noncommutative rational function and define a numerical invariant that measures its complexity. Using these results we determine concrete size bounds for rational identity testing, construct minimal symmetric realizations and prove an effective local-global principle for linear dependence of noncommutative rational functions.

Abstract. Consider a monic linear pencil L(x) = I − A 1 x 1 − · · · − A g x g whose coefficients A j are d×d matrices. It is naturally evaluated at g-tuples of matrices X using the Kronecker tensor product, which gives rise to its free locus Z (L) = {X : det L(X) = 0}. In this article it is shown that the algebras A and A generated by the coefficients of two linear pencils L and L, respectively, with equal free loci are isomorphic up to radical, i.e., A/ rad A ∼ = A/ rad A. Furthermore, Z (L) ⊆ Z ( L) if and only if the natural map sending the coefficients of L to the coefficients of L induces a homomorphism A/ rad A → A/ rad A. Since linear pencils are a key ingredient in studying noncommutative rational functions via realization theory, the above results lead to a characterization of all noncommutative rational functions with a given domain. Finally, a quantum version of Kippenhahn's conjecture on linear pencils is formulated and proved: if hermitian matrices A 1 , . . . , A g generate M d (C) as an algebra, then there exist hermitian matrices X 1 , . . . , X g such that i A i ⊗ X i has a simple eigenvalue.The arXiv version of the article contains an appendix presenting an invariant-theoretic viewpoint and is due independently to Claudio Procesi andŠpelaŠpenko.

We introduce a remarkable new family of norms on the space of n×n$n \times n$ complex matrices. These norms arise from the combinatorial properties of symmetric functions, and their construction and validation involve probability theory, partition combinatorics, and trace polynomials in non‐commuting variables. Our norms enjoy many desirable analytic and algebraic properties, such as an elegant determinantal interpretation and the ability to distinguish certain graphs that other matrix norms cannot. Furthermore, they give rise to new dimension‐independent tracial inequalities. Their potential merits further investigation.

Abstract. Call a noncommutative rational function Ö regular if it has no singularities, i.e., Ö(X) is defined for all tuples of self-adjoint matrices X. In this article regular noncommutative rational functions Ö are characterized via the properties of their (minimal size) linear systems realizations Ö = b * L −1 c. It is shown that Ö is regular if and only if L = A 0 + ∑ j A j x j is free elliptic. Roughly speaking, a linear pencil L is free elliptic if, after a finite sequence of basis changes and restrictions, the real part of A 0 is positive definite and the other A j are skew-adjoint. The second main result is a solution to a noncommutative version of Hilbert's 17th problem: a positive regular noncommutative rational function is a sum of squares.