A note on the limit points associated withthe generalized abc-conjecture for ℤ[t]

Daniel Davies

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A consequence of an effective form of the abc-conjecture

Jerzy Browkin (1999)

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T. Cochrane and R. E. Dressler [CD] proved that the abc-conjecture implies that, for every > 0, the gap between two consecutive numbers A $A^{0.4}$ with two exceptions given in Table 2.

Thomas’ conjecture over function fields

Volker Ziegler (2007)

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Thomas’ conjecture is, given monic polynomials $p_{1},$ $..., p_{d} \in ℤ [a]$ with $0 < deg p_{1} < \dots < deg p_{d}$ , then the Thue equation (over the rational integers) $(X - p_{1} (a) Y) \dots (X - p_{d} (a) Y) + Y^{d} = 1$ has only trivial solutions, provided $a \geq a_{0}$ with effective computable $a_{0}$ . We consider a function field analogue of Thomas’ conjecture in case of degree $d = 3$ . Moreover we find a counterexample to Thomas’ conjecture for $d = 3$ .

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The growth of regular functions on algebraic sets

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We are concerned with the set of all growth exponents of regular functions on an algebraic subset V of $ℂ^{n}$ . We show that its elements form an increasing sequence of rational numbers and we study the dependence of its structure on the geometric properties of V.