Solomon-Terao polynomials and Castelnouvo-Mumford regularity of hyperplane arrangements

The Solomon-Terao bi-polynomial was introduced by Solomon and Terao which degenerates to the characteristic polynomial of hyperplane arrangements. Also, it was proved recently that the other specialization of the Solomon-Terao bi-polynomial, we call the Solomon-Terao polynomial, coincides with the Poinrar\'{e} polynomial of the regular nilpotent Hessenberg variety when the arrangement and the variety comes from the same lower ideal in the positive system. Moreover, there are recent developments with superspace coinvariants and Fields conjecture, thus these polynomials are becoming more and more important. However, the research of them has been very hard, and even the top degree of the Solomon-Terao polynomial has not yet been known, which we solve in this article, by using the Castelnouvo-Mumford regularity of the logarithmic derivation modules.