TITLE: Polynomial bijection from ZxZ to Z?
QUESTION [26 upvotes]: It is known that the polynomial $f(n,m)=\frac{1}{2}(n+m)(n+m+1)+m$ defines bijection
$\mathbb{N}\times\mathbb{N}\to\mathbb{N}$ (Put pairs of $\mathbb{N}$ into the semi-infinite matrix and count them by diagonals). Does there exist a polynomial bijection 
$\mathbb{Z}\times\mathbb{Z}\to\mathbb{Z}$? The question is related to the open question 
about polynomial bijection $\mathbb{Q}\times\mathbb{Q}\to\mathbb{Q}$ here.

REPLY [10 votes]: It is an open problem. Maximal results about bijections from
$\mathbb N\times \mathbb N$, $\mathbb Z\times \mathbb N$, $\mathbb Z\times \mathbb Z$  to $\mathbb N$ are contained in

John S. Lew, Arnold L. Rosenberg,
  Polynomial indexing of integer lattice-points I. General concepts and quadratic polynomials, J. Number Theory 10 (1978) pp 192-214, doi:10.1016/0022-314X(78)90035-5.
Polynomial indexing of integer lattice-points II. Nonexistence results for higher-degree polynomials, J. Number Theory 10 (1978) pp 215-243,
  doi:10.1016/0022-314X(78)90036-7