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Let K be a number field, t a parameter, F = K(t), and ϕ(x) ∈ K[x] a polynomial of degree d ≥ 2. The polynomial Φ n (x, t) = ϕ • n (x)−t ∈ F [x], where ϕ • n = ϕ • ϕ • · · · • ϕ is the n-fold iterate of ϕ, is absolutely irreducible over F ; we compute a recursion for its discriminant. Let F ϕ be the field obtained by adjoining to F all roots (in a fixed F)… (More)

- Wayne Aitken, Michael D. Fried, Linda M. Holt
- 2003

We call a pair of polynomials f, g ∈ F q [T ] a Davenport pair (DP) if their value sets are equal, V f (F q t) = V g (F q t), for infinitely many extensions of F q. If they are equal for all extensions of F q (for all t ≥ 1), then we say (f, g) is a strong Davenport pair (SDP). Exceptional polynomials and SDP's are special cases of DP's.… (More)

- W. AITKEN, F. LEMMERMEYER
- 2005

After Hasse had found the first example of a Local-Global principle in the 1920s by showing that a quadratic form in n variables represented 0 in rational numbers if and only if it did so in every completion of the rationals, mathematicians investigated whether this principle held in other situations. Among the first counterexamples to the Hasse principle… (More)

Algorithmic logic is the logic of basic statements concerning algorithms and the algorithmic rules of deduction between such statements. It is a type-free logic capable of significant self-reference. Because of its expressive strength, traditional rules of logic are not necessarily valid. As shown in [1], the threat of paradoxes, such as the Curry paradox ,… (More)

- W. AITKEN, F. LEMMERMEYER
- 2005

We give an elementary, self-contained exposition concerning counterexamples to the Hasse Principle. Our account, which uses only techniques from standard undergraduate courses in number theory and algebra, focusses on counterexamples similar to the original ones discovered by Lind and Re-ichardt. As discussed in an appendix, this type of counterexample is… (More)

α-recursion lifts classical recursion theory from the first transfinite ordinal ω to an arbitrary admissible ordinal α [13]. Turing machine models for α-recursion and other types of transfinite computation have been proposed and studied [5] and [7] and are applicable in computational approaches to the foundations of logic and mathematics [11]. They also… (More)

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