Lattice point counts for the Shi arrangement and other affinographic hyperplane arrangements

摘要

Hyperplanes of the form x(j) = x(i) + c are called affinographic. For an affinographic hyperplane arrangement in R-n, such as the Shi arrangement, we study the function f (m) that counts integral points in [1, m](n) that do not lie in any hyperplane of the arrangement. We show that f (m) is a piecewise polynomial function of positive integers m, composed of terms that appear gradually as m increases. Our approach is to convert the problem to one of counting integral proper colorations of a rooted integral gain graph. An application is to interval coloring in which the interval of available colors for vertex vi has the form [h(i) + 1, m].