Powerful values of polynomials and a conjecture of Vojta

Pasten, Hector

doi:10.1016/j.jnt.2013.03.001

Cited by 13 publications

(27 citation statements)

References 21 publications

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“…Pasten also obtains results in logic in [Pas13]. He proves an undecidability result for systems of linear equations when some prescribed unknowns are required to be powers, again under Vojta's general ABC conjecture for algebraic numbers of bounded degree.…”

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confidence: 88%

“…The proof is similar to the proofs in Section 5 of [Pas13], but in our case we use Theorem 10 as our arithmetic input. Note that the analogous undecidability result for squares (rather than powers) is established under the Bombieri-Lang conjecture in [Voj00a].…”

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confidence: 96%

“…He also gives a solution for the analogue of Problem 1 for function fields over fields of characteristic zero using completely different methods (Nevanlinna theory), but his bounds on n are much weaker than ours due to the methods that he uses. Let us briefly compare the conjectural assumptions in our work and those in Pasten's work [Pas13]. The Bombieri-Lang conjecture and the 4-term ABC conjecture that we use concern a fixed number field, while Vojta's ABC conjecture used in [Pas13] concerns algebraic points of bounded degree in infinitely many number fields.…”

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confidence: 99%

“…Let us briefly compare the conjectural assumptions in our work and those in Pasten's work [Pas13]. The Bombieri-Lang conjecture and the 4-term ABC conjecture that we use concern a fixed number field, while Vojta's ABC conjecture used in [Pas13] concerns algebraic points of bounded degree in infinitely many number fields. This is necessary even if one is only interested in Q.…”

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confidence: 99%

“…The problem of finiteness of non-trivial sequences of length n formed by powers with second differences equal to 2 in number fields was conditionally solved by Pasten [Pas13] under Vojta's general ABC conjecture for algebraic numbers of bounded degree [Voj98]. He also gives a solution for the analogue of Problem 1 for function fields over fields of characteristic zero using completely different methods (Nevanlinna theory), but his bounds on n are much weaker than ours due to the methods that he uses.…”

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confidence: 99%

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Sequences of powers with second differences equal to two and hyperbolicity

Garcia-Fritz

2017

Trans. Amer. Math. Soc.

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By explicitly finding the complete set of curves of genus 0 or 1 in some surfaces of general type, we prove that under the Bombieri-Lang conjecture for surfaces, there exists an absolute bound M > 0 such that there are only finitely many sequences of length M formed by k-th rational powers with second differences equal to 2. Moreover, we prove the unconditional analogue of this result for function fields, with M depending only on the genus of the function field. We also find new examples of Brody-hyperbolic surfaces arising from the previous arithmetic problem. Finally, under the Bombieri-Lang conjecture and the ABC-conjecture for four terms, we prove analogous results for sequences of integer powers with possibly different exponents, in which case some exceptional sequences occur.

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confidence: 88%

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confidence: 96%

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confidence: 99%

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confidence: 99%

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confidence: 99%

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