Polynomial indexing of integer lattice-points II. Nonexistence results for higher-degree polynomials

Abstract

Denoting the nonnegative (resp. signed) integers by N (resp. Z) and the real numbers by R, let S ⊂ R 2 and f: R 2 → R. This current portion of our work specializes the basic concepts of Part I: Here f is a storing (resp. packing) function on S when f|( Z 2 ∩ S) is an injection into (resp. bijection onto) N; and the density S ÷ f is lim r→∞( 1 n ) # {Z 2 ∩ S ∩ f −1([−n, +n])} . Take a rational sector of the plane R 2 to mean a closed sector with rational boundary rays, and consider the plane itself as a rational sector. Define the function f to be sectorially increasing on a set S if this f is eventually increasing on some rational sectors S i constituting finite family which covers S. This paper obtains the following conclusions: A nonquadratic polynomial f( x, y) on a rational sector S with nonvoid interior, provided either f is sectorially increasing or deg( f) ≤ 4, cannot be a storing function with unit density on the sector S. Our proofs of these conclusions involve a strong criterion for zero density. These results support a nonexistence conjecture in Part I of our work, and extend an earlier remark of Pólya and Szegö. Indeed these results, under any one of our auxiliary assumptions, exclude nonquadratic storing functions with unit density on the arrays Z 2, Z × N, N 2.