The Smith Normal Form Of Hyperplane Arrangement

The determinant theorem of the Varchenko matrix is proposed by A.Varchenko,it is closely related to the Smith normal form and the hyperplane theory.R P. Stanley proposed a problem that whether the Varchenko matrix of arbitrary hyperplane arrangement on n-dimensional vector space have Smith normal form on the basis of Varchenko theory.This paper on the basis of R P. Stanley’s problem study the Smith normal form of q-Varchenko matrix of some special hyperplane arrangement, including peelable hyperplane arrangement on n-dimensional vector space,hyperplane arrangement Cn that the lines through the same point in R2,and regular n-gon arrangement Pn in R2 which is extended by the edge of the regular n-gon.This paper prove the existence of the Smith normal of these hyperplane arrangement,and get formula of Cn and Pn arrangement.The main result of this paper is the solution of the Smith normal form of the Pn arrangement.For the calculation of P7 arrangement and the result of P3 up to P6,this paper propose a problem to find the formula of Smith normal of Pn.This problem is solved by using some way to classify the regions for calculation successfully.