8 Main results

8.1 Proposition 1

Proposition 37 Open Bruhat cell \(=\) nonvanishing minors

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The open Bruhat cell equals the set of matrices with nonvanishing leading principal minors. Equivalently, the values \(z_1^*,\ldots ,z_r^*\) are soluble if and only if all leading principal minors are nonzero.

Proof ▶

By definition of the open Bruhat cell.

8.2 The residue formula

Theorem 38 Intermediate residue formula

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Under convergence hypotheses, the integral equals \((2\pi i)^n\) times the sum of flag residues over arising selections.

Proof ▶

Rewrite the integrand via ofRealForm, apply UHP-filtered expansion, and translate via the arising equivalence.

Theorem 39 Arising sum equals stable sum

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On the generic locus (for every selection whose coefficient matrix is invertible, the spectator product is nonzero), the sum of flag residues over arising selections equals the sum over \(\Pi \)-stable selections. Selections with singular coefficient matrix are inert: no ordering of them arises or is \(\Pi \)-stable.

Proof ▶

Apply the cancellation identity \(\chi = \) stable sum, after identifying the arising signed sum with \(\chi \) via Cramer’s rule.

Theorem 40 Main Theorem: Residue formula for \(\Pi \)-stable selections

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Let \(s_j\) have positive real part. Under the convergence hypothesis and on the generic locus (for every selection whose coefficient matrix is invertible, the spectator product is nonzero),

\[ \int _{\mathbb {R}^n} \frac{h(x)}{\prod _{j=1}^R (f_j(x) - s_j\, i)}\, dx = (2\pi i)^n \sum _{\sigma \in \Pi \text{-stable}} \mathrm{flagResidue}(\sigma ). \]

The sum runs only over \(\Pi \)-stable selections.

Proof ▶

Combine the intermediate residue formula with the equality of the arising and stable sums (the cancellation argument).