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Null- and Positivstellensätze for rationally resolvable ideals
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Null- and Positivstellensätze for rationally resolvable ideals

Igor Klep, Victor Vinnikov and Jurij Volčič
Linear algebra and its applications, v 527, pp 260-293
15 Aug 2017
DOI: https://doi.org/10.1016/j.laa.2017.04.009
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https://doi.org/10.1016/j.laa.2017.04.009View
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Abstract

Division ring Free algebra Free analysis nc unitary group Nullstellensatz Positivstellensatz Rational identity Real algebraic geometry Skew field Spherical isometry
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.

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