Parametrizing Over ℤ Integral Values of Polynomials Over ℚ

Given a polynomial f ∈ ℚ[X] such that f(ℤ) ⊂ ℤ, we investigate whether the set f(ℤ) can be parametrized by a multivariate polynomial with integer coefficients, that is, the existence of g ∈ ℤ[X 1,…, X m ] such that f(ℤ) = g(ℤ m ). We offer a necessary and sufficient condition on f for this to be possible. In particular, it turns out that some power of 2 is a common denominator of the coefficients of f, and there exists a rational β with odd numerator and odd prime-power denominator such that f(X) = f(β −X). Moreover, if f(ℤ) is likewise parametrizable, then this can be done by a polynomial in one or two variables.