A hyperplane arrangement A is a finite collection of hyperplanes in a vector space. People are interested in the geometric, topological, algebraic, and combinatorial properties of its complement, which is the remaining set after removing all members of A from the whole space.

The most important invariant associated with a hyperplane arrangement is the characteristic polynomial. In this talk, we shall present, after a brief introduction, some new combinatorial phenomenon on this polynomial, including the alternating partial sum of its coefficients, its reciprocity theorem, its translation deformation, and the coefficient comparison on its quasi-polynomial.