### The diophantine theory of a ring of analytic functions.

Skip to main content (access key 's'),
Skip to navigation (access key 'n'),
Accessibility information (access key '0')

Back to Simple Search
# Advanced Search

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 $K$ has a negative answer if $K$ satisfies two arithmetical conditions (existence of a so-called set of integers and of an elliptic curve of rank one over $K$). We relate division-ample sets to arithmetic of abelian varieties.

**Page 1**