Remarks on polynomial parametrization of sets of integer points

Details Remarks on polynomial parametrization of sets of integer points Remarks on polynomial parametrization of sets of integer points

2007-06-03

arxiv.org/abs/0706.0292v2

Communications in Algebra 36 (2008) (3) 1110 - 1114

Mathematics

Number Theory

to appear in Comm. Algebra

Scientific paper

10.1080/00927870701776938

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.

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