5 Selections and arising cones
An \(n\)-element selection \(\sigma : \mathrm{Fin}\, n \hookrightarrow \mathrm{Fin}\, R\) gives rise to a combined polar data.
A selection \(\sigma \) is \(\Pi \)-stable.
The finite set of all \(\Pi \)-stable embeddings \(\mathrm{Fin}\, n \hookrightarrow \mathrm{Fin}\, R\).
The arising cone for \(\sigma \): the set of real parameter vectors where \(\sigma \) arises (i.e., \(z_k^*\) is in the upper half-plane at each step).
The recursive \(\mathrm{poleIn0}\)-based arising predicate is equivalent to the \(z^*\)-based arising predicate. Proved using the Schur complement identities.
Proof
Induction on dimension. The Schur complement identities show that the imaginary numerator ratio is preserved under restriction.