Skolem functions of arithmetical sentences

Arithmetical formulas are the formulas containing the usual logical and arithmetical symbols +,., and constants of Z. If an arithmetical sentence ?x?y ψ (x, y) is true in a model M, then there is function f(x) defined on M such that ?x ψ (x, f(x)) is true in M. Such a function is called a Skolem function of the arithmetical sentence ?x?y ψ(x, y). In this paper, we study the bounds of the Skolem functions when the model M is the set of all natural numbers N or the ring of integers Z. We define the Skolem function f(x) for ?x?y ψ(x, y) as follows. For any a in N (or Z) let f(a) be the least (or least absolute value of) b such that ψ(a, b) is true in N (or Z). For every arithmetical sentence ?x?y?z ψ(x, y, z) true in N (or Z) there is a polynomial g(x) over Z such that the corresponding Skolem function f(x) < g(|x|) for any x in N (or Z). An application of considering the bounds of these Skolem function is the following: If the Generalized Riemann Hypothesis holds, then for every d there is a polynomial time algorithm for the following problem: given a quantifier-free arithmetical formula φ(x, y) of degree at most d, does ?x?y φ(x, y) hold in Z. Moreover, if the sentence is false in Z, then the algorithm outputs an a ∈ Z such that ?y ¬ φ(a, y).