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We prove the addition-deletion theorems for the Solomon-Terao polynomials,which have two important specializations. Namely, one is to the characteristicpolynomials of hyperplane arangements, and the other to the Poincar\`{e}polynomials of the regular nilpotent Hessenberg varieties. One of the maintools to show them is the free surjection theorem which confirms the rightexactness of several important exact sequences among logarithmic modules.Moreover, we introduce a generalized polynomial $B$-theory to the higher orderlogarithmic modules, whose origin was due to Terao.

Addition-deletion theorems for the Solomon-Terao polynomials and $B$-sequences of hyperplane arrangements