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<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 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<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−∑_jX_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.