Remarks on polynomial parametrization of sets of integer points. (English) Zbl 1209.11038

Frisch, Sophie

Commun. Algebra 36, No. 3, 1110-1114 (2008).

Summary: If, for a subset \(S\) of \(\mathbb Z^{k}\), 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 \(\Rightarrow \) b (obviously), b \(\Rightarrow\) 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 \(\mathbb Z^{k}\) is parametrizable a single \(k\)-tuple of polynomials with integer coefficients.

MSC:

11D85	Representation problems
11C08	Polynomials in number theory
13F20	Polynomial rings and ideals; rings of integer-valued polynomials

Keywords:

image of a polynomial; integer-valued polynomial; polynomial mapping; polynomial parametrization; range

References:

[1]	Cahen P.-J., Integer-Valued Polynomials (1997)
[2]	Frisch S., J. Pure Appl. Algebra 212 pp 271– (2008) · Zbl 1215.11025
[3]	Nathanson M. B., Additive Number Theory. The Classical Bases (1996) · Zbl 0859.11002