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arXiv · 1504.08004

Null- and Positivstellens\"atze for rationally resolvable ideals

Igor Klep·Victor Vinnikov·Jurij Volčič
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Abstract

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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Igor Klep, Victor Vinnikov, Jurij Volčič. 2015-04-29. Null- and Positivstellens\"atze for rationally resolvable ideals. https://doi.org/10.1016/j.laa.2017.04.009

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