Does there exist an algorithm which to each Diophantine equation assigns an integer which is greater than the number (heights) of integer solutions, if these solutions form a finite set?

Apoloniusz Tyszka

Abstract

Let En = {xi = 1, xi + xj = xk , xi · xj = xk : i, j, k ∈ {1, . . . , n}}. If Matiyasevich’s conjecture on finite-fold Diophantine representations is true, then for every computable function f : N → N there is a positive integer m(f) such that for each integer n ≥ m(f) there exists a system S ⊆ En which has at least f(n) and at most finitely many solutions in integers x1, . . . , xn. This conclusion contradicts to the author’s conjecture on integer arithmetic, which implies that the heights of integer solutions to a Diophantine equation are computably bounded, if these solutions form a finite set.
Author Apoloniusz Tyszka (FoPaPE)
Apoloniusz Tyszka,,
- Faculty of Production and Power Engineering
Journal seriesFundamenta Informaticae, ISSN 0169-2968, (A 20 pkt)
Issue year2013
No125
Pages95-99
Publication size in sheets0.5
Keywords in EnglishDavis-Putnam-Robinson-Matiyasevich theorem, Matiyasevich's conjecture on finite-fold Diophantine representations
DOIDOI:10.3233/FI-2013-854
URL http://arxiv.org/pdf/1105.5747v27.pdf
Internal identifierWIPiE/2013/109
Languageen angielski
Score (nominal)20
ScoreMinisterial score = 15.0, 26-07-2017, ArticleFromJournal
Ministerial score (2013-2016) = 20.0, 26-07-2017, ArticleFromJournal
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