Let R denote the ring Q or Z. (Not the reals!) Then R^n denotes
the cartesian product R x ... x R (n times).
Suppose f : R^n —> R^n is a polynomial mapping of the form
f(x_1, ..., x_n) = (P_1(x_1,...,x_n), ..., P_n(x_1,...,x_n))
where each P_j is a polynomial in the x_1, ..., x_n over the ring R.
Question:
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(*) If f is a bijection, does that imply that each P_j is linear ?
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The polynomial bijections of R^n —> R^n (R = Q or Z) form a group.
Just in case there are any non-linear ones: What is this group?
E.g. for Z^2 or Q^2.
It is known that if the ring is the complex field C, there do exist
non-linear polynomial bijections C^2 —> C^2 of form
f(z,w) = (P(z,w), Q(z,w)).
Of course the same question (*) can be asked about any ring at all.
—Dan
