On the symmetry of \(b\)-functions of linear free divisors
The authors introduce the concept of a prehomogeneous determinant as a generalization for linear free divisors. They show that the \(b-\)function roots are symmetric about \(-1\) for reductive prehomogeneous determinants and for the regular special free divisor. As a consequence, they get a positive answer to a question raised by Castro-Jiménez and Ucha-Enriquez, saying that \(-1\) is the only integer root of the corresponding \(b-\)function. As an application, the authors obtain new cases when the logarithmic comparison theorem holds.
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