An Algorithm For Characteristic Polynomials Of Hyperplane Arrangements

In this thesis, the definition of coefficient matrix for a general hyperplane arrangement is given, and optimize the characteristic matrix for a general hyperplane arrangement. According to the Whitney's theorem, calculating of characteristic matrix can be done by calculating the rank of the sub-matrices of the coefficient matrix. The algorithm of characteristic polynomial of an arrangement is formulated. A classification of arrangements with at most seven lines in 2-dimensional space and five planes in 3-dimensional space was made under the characteristic matrix. Arrangements with two lines can be divided into 2 classes, arrangements with three lines can be divided into 4 classes, arrangements with four lines can be divided into 8 classes, arrangements with five lines can be divided into 17 classes, arrangements with six lines can be divided into 37 classes, arrangements with seven lines can be divided into 78 classes. Arrangements with two planes can be divided into 2 classes, arrangements with three planes can be divided into 5 classes, arrangements with four planes can be divided into 12 classes, arrangements with five planes can be divided into 43 classes.At last, some applications in topological classification of polymers are given.