We prove the addition-deletion theorems for the Solomon-Terao polynomials, which have two important specializations. Namely, one is to the characteristic polynomials of hyperplane arangements, and the other to the Poincarè polynomials of the regular nilpotent Hessenberg varieties. One of the main tools to show them is the free surjection theorem which confirms the right exactness of several important exact sequences among logarithmic modules. Moreover, we introduce a generalized polynomial $B$-theory to the higher order logarithmic modules, whose origin was due to Terao.