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\title[Affine semicircular determinants]{A variational formula for affine semicircular determinants}
\author{Sheng Yin}
\address{Institute for Advanced Study in Mathematics, Harbin Institute of Technology}
\email{sheng.yin@hit.edu.cn}
\date{October 10, 2026}
\begin{document}
\begin{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.
\end{abstract}
\maketitle
\tableofcontents
\input{sections/introduction}
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\input{sections/positive-pencils}
\input{sections/ordered-minmax}
\input{sections/spectral}
\appendix
\input{sections/comparison-appendix}
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\end{document}