A Property of the Roots of \(r\)-D Rook Polynomials

Abstract

In this paper, we prove an interesting property of rook polynomials for \(2\)-D square boards and extend that for rook polynomials for \(3\)-D cubic, and \(r\)-D “hypercubic” boards. In particular, we prove that for \(r\)-D rook polynomials the modulus of the sum of their roots equals their degree. We end with some further questions, mainly for the \(2\)-D and \(3\)-D case, that could serve as future projects.