On a generalization of Solomon-Terao formula for subspace arrangements

TL;DR

This paper extends the Solomon-Terao formula to central equidimensional subspace arrangements by introducing generalized Solomon-Terao functions and establishing their polynomial nature and relation to characteristic polynomials.

Contribution

It generalizes the Solomon-Terao formula to subspace arrangements using modules of multi-logarithmic forms and residues, and proves its validity for line arrangements of any codimension.

Findings

Generalized Solomon-Terao functions are polynomial.

Relation established between Solomon-Terao polynomial and characteristic polynomial.

Formula holds for any line arrangement of any codimension.

Abstract

We investigate in this paper a generalization of Solomon-Terao formula for central equidimensional subspace arrangements. We introduce generalized Solomon-Terao functions based on the Hilbert-Poincar\'e series of the modules of multi-logarithmic forms and logarithmic multi-residues. We show that as in the case of hyperplane arrangements, these Solomon-Terao functions are polynomial. We then prove that if the Solomon-Terao polynomial of the modules of multi-residues satisfies a certain property, then this polynomial is related to the characteristic polynomial of the subspace arrangement. In particular, we prove that this generalized Solomon-Terao formula holds for any line arrangement of any codimension.