Jurij Volčič

On domains of noncommutative rational functions

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.

Null- and Positivstellensätze for rationally resolvable ideals

Abstract Hilberts Nullstellensatz characterizes polynomials that vanish on the vanishing set of an ideal in C [ X _ ] . In the free algebra C X _ > the vanishing set of a two-sided ideal I is defined in a dimension-free way using images in finite-dimensional representations of C X _ > / I . In this article Nullstellensatze for a simple but important class of ideals in the free algebra – called tentatively rationally resolvable here – are presented. An ideal is rationally resolvable if its defining relations can be eliminated by expressing some of the X _ variables using noncommutative rational functions in the remaining variables. Whether such an ideal I satisfies the Nullstellensatz is intimately related to embeddability of C X _ > / I into (free) skew fields. These notions are also extended to free algebras with involution. For instance, it is proved that a polynomial vanishes on all tuples of spherical isometries iff it is a member of the two-sided ideal I generated by 1 − ∑ j X j ⊺ X j . This is then applied to free real algebraic geometry: polynomials positive semidefinite on spherical isometries are sums of Hermitian squares modulo I . Similar results are obtained for nc unitary groups.

Regular and positive noncommutative rational functions

Call a noncommutative rational function

Positive trace polynomials and the universal Procesi–Schacher conjecture

r

Skolem–Noether algebras

regular if it has no singularities, i.e.,

Geometry of free loci and factorization of noncommutative polynomials

r(X)

Matrix coefficient realization theory of noncommutative rational functions

is defined for all tuples of self-adjoint matrices

Free loci of matrix pencils and domains of noncommutative rational functions

X

Stable noncommutative polynomials and their determinantal representations.

. In this article regular noncommutative rational functions

Noncommutative polynomials describing convex sets.

r