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A hypothetical upper bound for solutions of a Diophantine equation with a finite number of solutions.

Пнд, 08/17/2009 - 18:30 — SomNambul
Authors: Apoloniusz Tyszka
Subjects: Number Theory
link: http://arxiv.org/abs/0901.2093
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}}, K \in
{Z,Q,R,C}. We conjecture that if a system S \subseteq E_n has only finitely
many solutions in K, then their number does not exceed 2^n. We prove this bound
for K=C. We construct a system S \subseteq E_{21} such that S has infinitely
many integer solutions and S has no integer solution in
[-2^{2^{21-1}},2^{2^{21-1}}]^{21}. We conjecture that if a system S \subseteq
E_n has a finite number of solutions in K, then each such solution
(x_1,...,x_n) satisfies (|x_1|,...,|x_n|) \in [0,2^{2^{n-1}}]^n. Applying this
conjecture for K=Z, we prove that if a Diophantine equation has only finitely
many integer (rational) solutions, then their heights are bounded from above by
a constant which recursively depends on the degree and the coefficients of the
equation. We note that an affirmative answer to the famous open problem whether
each listable set M \subseteq Z^n has a finite-fold Diophantine representation
would falsify our conjecture for K=Z.

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