Remarks on Polynomial Parametrization of Sets of Integer Points

Abstract

If, for a subset S of ℤ^k, we compare the conditions of being parametrizable by (a) a single k-tuple of polynomials with integer coefficients, (b) a single k-tuple of integer-valued polynomials, and (c) finitely many k-tuples of polynomials with integer coefficients (variables ranging through the integers in each case), then a ⇒ b (obviously), b ⇒ c, and neither implication is reversible. Condition (b) is equivalent to S being the set of integer k-tuples 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 parametrizable a single k-tuple of polynomials with integer coefficients.

Key Words:

• Image of a polynomial
• Integer-valued polynomial
• Polynomial mapping
• Polynomial parametrization
• Range

2000 Mathematics Subject Classification:

• Primary 11D85
• Secondary 11C08, 13F20

Notes

Communicated by L. Ein.

This note was written while the author was enjoying hospitality at Université de Picardie, Amiens.