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?
Abstract
Let E_n={x_i=1, x_i+x_j=x_k, x_i \cdot x_j=x_k: i,j,k \in {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 \subseteq E_n which has at least f(n) and at most finitely many solutions in integers x_1,...,x_n. 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.
Keywords
Cite
@article{arxiv.1105.5747,
title = {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?},
author = {Apoloniusz Tyszka},
journal= {arXiv preprint arXiv:1105.5747},
year = {2014}
}
Comments
Unchanged text, the conjecture with the bound 2^(2^(n-1)) is false, see http://dx.doi.org/10.13140/2.1.1707.2640 arXiv admin note: text overlap with arXiv:0901.2093, arXiv:1102.4122, arXiv:1011.4103