Remarks on Polynomial Parametrization of Sets of Integer Points

If, for a subset S of Z, 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 ⇒ b (obviously), b ⇒ c, and neither implication is reversible. We give different characterizations of condition (b). Also, we show that every co-finite subset of Z is parametrizable a single k-tuple of polynomials with integer coefficients. If f = (f1, . . . , fk) ∈ (Z[x1, . . . , xn]) k is a k-tuple of polynomials with integer coefficients in several variables, we call range or image of f the range of the function f :Z −→ Z defined by substitution of integers for the variables; and likewise for a k-tuple of integer-valued polynomials (f1, . . . , fk) ∈ (Int(Z )), where Int(Z) = {g ∈ Q[x1, . . . , xn] | ∀a ∈ Z n : g(a) ∈ Z}. If S ⊆ Z is the range of f = (f1, . . . , fk), we say that f parametrizes S. We want to compare two kinds of polynomial parmetrization of sets of integers or k-tuples of integers: by integer-valued polynomials and by polynomials with integer coefficients. Consider for instance the set of integer Pythagorean triples: it takes two triples of polynomials with integer coefficients, ( c(a − b), 2cab, c(a + b) )