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A variational formula for affine semicircular determinants

Primarily AI-generated textHuman understanding: no partsmath.OA — Operator Algebrasmath.PR — Probability

Contributed by Sheng Yin ↗

Version 1 / Oct 10, 2026 / CC0 1.0

Abstract

We express the Fuglede--Kadison determinant of an affine pencil in free semicircular operators as an infimum over positive definite matrices. The infimum is attained if and only if the pencil is boundedly invertible, in which case its minimizer is unique. At zero constant term, a scalar rescaling recovers the determinant-capacity identity of Mai and Speicher. For selfadjoint coefficients we compare the formula with a strictly constrained Hilbert--Schmidt variational principle and establish ordered min--max formulas under their respective positivity hypotheses. Strictly positive pencils admit a minimum over a linear matrix inequality. Applications to Brown measure include a uniform determinant lower bound, restrictions on atoms, radial distribution formulas, and a Hessian formula for the regularized density. The text of this manuscript was generated entirely by OpenAI's large language model Astra.

Provenance statement

The mathematical content and text of this manuscript were generated entirely by an autonomous exploration system driven by OpenAI’s large language model Astra under human guidance.

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Version history

  1. v1Submitted by Sheng YinInitial depositCurrentOct 10, 2026