PARAMETRIZING OVER Z INTEGRAL VALUES OF POLYNOMIALS OVER Q

摘要

Given a polynomial f is an element of Q[X] such that f(Z) subset of Z, we investigate whether the set f(Z) can be parametrized by a multivariate polynomial with integer coefficients, that is, the existence of g is an element of[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 beta with odd numerator and odd prime-power denominator such that f(X) = f(beta-X). Moreover, if f(Z) is likewise parametrizable, then this can be done by a polynomial in one or two variables.