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.