Packing polynomials on multidimensional integer sectors (Q727172)

Summary: Denoting the real numbers and the nonnegative integers, respectively, by \(R\) and \( N\), let \(S\) be a subset of \( N^n\) for \(n = 1, 2,\ldots\), and \(f\) be a mapping from \( R^n\) into \( R\). We call \(f\) a packing function on \(S\) if the restriction \(f|_{S}\) is a bijection onto \(N\). For all positive integers \(r_1,\ldots,r_{n-1}\), we consider the integer sector \(I(r_1, \ldots, r_{n-1}) =\{(x_1,\ldots,x_n) N^n\mid x_{i+1}\leqslant r_i x_i \,\,\text{for}\,\, i = 1,\ldots,n-1 \}\). Recently, \textit{M. B. Nathanson} [J. Algebra Appl. 13, No. 5, Article ID 1350165, 14 p. (2014; Zbl 1290.05010)] 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 \( N^n\) we construct a packing polynomial on the sector \(I(1, \ldots, 1)\).