The Study On Characteristic Polynomials Of Generic Arrangements

This paper mainly studies a kind of special hyperplane arrangements,that is,generic arrangements.In the book Arrangements of Hyperplanes,the authors P.Orlik and H.Terao introduce the generic arrangements and give a conclusion.A central arrangement is generic if and only if it is a conical arrangement of a general position arrangement.R.P.Stanley also gives the definition of generic arrangements,which removes the restriction that the generic arrangements are the central arrangements,the definition is more general.In this paper,we study the generic arrangements given by R.P.Stanley,and we give the calculation method of the characteristic polynomials of the generic arrangements and their sub arrangements.The definition of the generalized general position arrangements is given,and we study the relationships among the general position arrangements,generalized general position arrangements and generic arrangements.In addition,by establishing the relationships between arrangements and the simple graphs,we study the sufficient and necessary condition for linear independence of the generic threshold arrangements and their subarrangements,and obtain the characteristic polynomials of the generic threshold arrangements and their subarrangements.Finally,we give the definition and related theorems of the semigeneric arrangements of type D_n.The main conclusions of this paper are as follows:In this paper,we first generalize the general position arrangements,give the definition of the generalized general position arrangements,and determine the relationships among the general position arrangements,generalized general position arrangements and generic arrangements.It is concluded that there are three sets of the general position arrangements,generalized general position arrangements and generic arrangements,and the former is the true subset of the latter.Because the generalized general position arrangements are the generic arrangements,with the help of the characteristic polynomials of generic arrangements,the characteristic polynomials of generalized general position arrangements are calculated.In addition,we study the deformation forms of the threshold arrangements which is called the generic threshold arrangements.By establishing the relationships between linear independent sub arrangements and simple graphs without even cycles,the characteristic polynomials of the generic threshold arrangements and their subarrangements are given.Finally,we study the semigeneric arrangements of type D_n.The relationships between the semigeneric arrangements of type D_n and labeled graphs are established,which make it easier to calculate the characteristic polynomials of the semigeneric arrangement of type D_n.