The diophantine theory of a ring of analytic functions.
Hilbert's Tenth Problem for fields of rational functions over finite fields.
Diophantine undecidability for addition and divisibility in polynomial rings
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.
Extensions of Büchi's problem: Questions of decidability for addition and kth powers
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...
Division-ample sets and the Diophantine problem for rings of integers
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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