Does a polynomial in two variables which establishes a bijection between

Question

Does a polynomial in two variables which establishes a bijection between the points with nonnegative integer coordinates and natural numbers exist? porve it

i think the anser is P(x,y)=x+y. but i dont know how to prove it or if i get the anser. — heyThere, May 19 '16 at 08:54

score 1 · Answer 1 · edited Apr 13 '17 at 12:20

1

A quadratic bijection is $$f(x, y) = \frac{(x+y)(x+y+1)}{2} + x$$ which is called the 'Cantor pairing function' (see this).

This probably answers your question, but for more information, you can see any of these answers (incidentally all by the same user).

edited Apr 13 '17 at 12:20

Community

1

answered May 19 '16 at 09:10

shardulc

4,562

score 0 · Accepted Answer · answered May 19 '16 at 09:01

The standard "zig-zag" bijection $\mathbb N\times \mathbb N\to \mathbb N$ (which corresponds to enumerating the pairs in order of increasing sum, and between pairs with the same sum in order of the first element) is polynomial.

I will leave it to you to derive the precise coefficients (which will depend on whether $0\in\mathbb N$ for you anyway) -- you will need to use the formula for the triangular numbers $\frac{n(n+1)}2$, which can give you the number of pairs with a smaller sum than the one you're looking at.

Does a polynomial in two variables which establishes a bijection between

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