Computational complexities of diophantine equations with parameters
Journal of Algorithms
Factoring multivariate polynomials over algebraic number fields
SIAM Journal on Computing
The bounds of Skolem functions and their applications
Information and Computation
Computational complexity of arithmetical sentences
Information and Computation
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
The Design and Analysis of Computer Algorithms
The Design and Analysis of Computer Algorithms
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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).