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We prove that the positive-existential theory of addition and divisibility in a ring of polynomials in two variables A[t₁,t₂] over an integral domain A is undecidable and that the universal-existential theory of A[t₁] is undecidable.
We generalize a question of Büchi: Let R be an integral domain, C a subring and k ≥ 2 an integer. Is there an algorithm to decide the solvability in R of any given system of polynomial equations, each of which is linear in the kth powers of the unknowns, with coefficients in C?
We state a number-theoretical problem, depending on k, a positive answer to which would imply a negative answer to the question for R = C = ℤ.
We reduce a negative answer for k = 2 and for...
We prove that Hilbert’s Tenth Problem for a ring of integers in a number field has a negative answer if satisfies two arithmetical conditions (existence of a so-called set of integers and of an elliptic curve of rank one over ). We relate division-ample sets to arithmetic of abelian varieties.
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