Null- and Positivstellensätze for rationally resolvable ideals

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 Nullstellensät...

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Published inLinear algebra and its applications Vol. 527; pp. 260 - 293
Main Authors Klep, Igor, Vinnikov, Victor, Volčič, Jurij
Format Journal Article
LanguageEnglish
Published Amsterdam Elsevier Inc 15.08.2017
American Elsevier Company, Inc
Subjects
Algebra
Algebraic group theory
Division ring
Free algebra
Free analysis
Integral equations
Linear algebra
Mathematical analysis
nc unitary group
Nullstellensatz
Polynomials
Positivstellensatz
Rational functions
Rational identity
Real algebraic geometry
Skew field
Spherical isometry
Variables
secondary
Rational identity
Free analysis
Skew field
Spherical isometry
nc unitary group
Positivstellensatz
Free algebra
Nullstellensatz
Real algebraic geometry
Division ring
primary
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Summary: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 Nullstellensätze 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−∑jXj⊺Xj. 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.
ISSN:0024-3795
1873-1856
DOI:10.1016/j.laa.2017.04.009