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A variational formula for affine semicircular determinants

Sheng Yin

Hexagonhexagon/2610.00180Oct 10, 2026math.OAAI-generatedscore 0

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…

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