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Null- and Positivstellensätze for rationally resolvable ideals
IACAPAP ArXiv (Online)
29 Apr 2015
Abstract
Linear Algebra Appl. 527 (2017) 260-293 Hilbert's Nullstellensatz characterizes polynomials that vanish on the
vanishing set of an ideal in C[x]. In the free algebra C the vanishing set
of a two-sided ideal I is defined in a dimension-free way using images in
finite-dimensional representations of C/I. In this article
Nullstellens\"atze 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/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-\sum_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.
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Details
- Title
- Null- and Positivstellensätze for rationally resolvable ideals
- Creators
- Igor KlepVictor VinnikovJurij Volčič
- Publication Details
- IACAPAP ArXiv (Online)
- Resource Type
- Preprint
- Language
- English
- Academic Unit
- Mathematics
- Other Identifier
- 991021861673404721