Abstract: We compare parametrizability by integer-valued polynomials and parametrizability by polynomials with integer coefficients. Every subset of Z^k that is representable as the range of a k-tuple of integer-valued polynomials is representable as a finite union of ranges of k-tuples of polynomials with integer coefficients (in the same number of variables), but the converse does not hold.
Also, there are sets parametrizable by a single k-tuple of integer-valued polynomials, that are not parametrizable by a single k-tuple of polynomials with integer coefficients (in any number of variables), for instance, the set of integer Pythagorean triples.
A different characterization of sets of k-tuples of integers that are parametrizable by integer-valued polynomials: they are precisely the sets that occur as the set of integer points in the range of a k-tuple of polynomials with rational coefficients, as the variables range through the integers.
Also, we show that every co-finite subset of Z^k is representable as
the range of a k-tuple of polynomials (f_1,...,f_k) in several
variables with integer coefficients.
2000 Mathematics Subject Classification: Primary 11D85; Secondary 13F20, 11C08.
This note was written while the author was enjoying hospitality at Université de Picardie, Amiens, in February 2006PDF: S. Frisch: ... polynomial parametrization of sets of integer points