Remarks on polynomial parametrization of sets of integer points

TL;DR

This paper explores different polynomial parametrizations of integer point sets, establishing implications among them and characterizing when sets are parametrizable by integer-valued polynomials or finitely many polynomial tuples.

Contribution

It clarifies the relationships between various polynomial parametrization conditions and characterizes co-finite sets as parametrizable by a single polynomial tuple.

Findings

Condition (a) implies (b), which implies (c).

Condition (b) is equivalent to being the set of integer values of a rational-coefficient polynomial tuple.

Every co-finite subset of Z^k is parametrizable by a single polynomial tuple.

Abstract

If, for a subset S of Z^k, we compare the conditions of being parametrizable (a) by a single k-tuple of polynomials with integer coefficients, (b) by a single k-tuple of integer-valued polynomials and, (c) by finitely many k-tuples of polynomials with integer coefficients (variables ranging through the integers in each case) then (a) implies (b) (obviously), (b) implies (c), and neither converse holds. Condition (b) is equivalent to the set S being the set of integer values taken by some k-tuple of polynomials with rational coefficients as the variables range through the integers. We also show that every co-finite subset of Z^k is parametrizable a single k-tuple of polynomials with integer coefficients.