Packing Polynomials on Multidimensional Integer Sectors

Abstract

Denoting the real numbers and the nonnegative integers, respectively, by ${\bf R}$ and ${\bf N}$, let $S$ be a subset of ${\bf N}^n$ for $n = 1, 2,\ldots$, and $f$ be a mapping from ${\bf R}^n$ into ${\bf R}$. We call $f$ a packing function on $S$ if the restriction $f|_{S}$ is a bijection onto ${\bf N}$. For all positive integers $r_1,\ldots,r_{n-1}$, we consider the integer sector \[I(r_1, \ldots, r_{n-1}) =\{(x_1,\ldots,x_n) \in N^n \; | \; x_{i+1} \leq  r_ix_i \mbox{ for } i = 1,\ldots,n-1 \}.\] Recently, Melvyn B. Nathanson (2014) proved that for $n=2$ there exist two quadratic packing polynomials on the sector $I(r)$. Here, for $n>2$ we construct $2^{n-1}$ packing polynomials on multidimensional integer sectors. In particular, for each packing polynomial on ${\bf N}^n$ we construct a packing polynomial on the sector $I(1, \ldots, 1)$.