Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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?   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.

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The categories of this paper are: stat
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The categories of this paper are: q-fin
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The categories of this paper are: supr-con
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The categories of this paper are: econ
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The categories of this paper are: nlin
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The categories of this paper are: atom-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: plasm-ph
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The categories of this paper are: cond-mat
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The categories of this paper are: alg-geom
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The categories of this paper are: math-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: ao-sci
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The categories of this paper are: quant-ph
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The categories of this paper are: adap-org
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The categories of this paper are: nucl-th