The diagonal polynomials of dimension four

The characterization of all bijective polynomials from Nn to N (packing polynomials of dimension n) is a difficult unsolved problem. Apparently a more tractable problem is the determination of diagonal polynomials, a subset of packing polynomials. However for this later problem, it is only known that dimension two admits just one normalized diagonal polynomial (precisely the Cantor polynomial), and dimension three admits just two. Here, we prove that dimension four admits six normalized diagonal polynomials (normalized polynomials determine all diagonal polynomials).