1,054 research outputs found
Diophantine Undecidability of Holomorphy Rings of Function Fields of Characteristic 0
Let be a one-variable function field over a field of constants of
characteristic 0. Let be a holomorphy subring of , not equal to . We
prove the following undecidability results for : If is recursive, then
Hilbert's Tenth Problem is undecidable in . In general, there exist
such that there is no algorithm to tell whether a
polynomial equation with coefficients in \Q(x_1,...,x_n) has solutions in
.Comment: This version contains minor revisions and will appear in Annales de l
Institut Fourie
Undecidability in number theory
These lecture notes cover classical undecidability results in number theory,
Hilbert's 10th problem and recent developments around it, also for rings other
than the integers. It also contains a sketch of the authors result that the
integers are universally definable in the rationals.Comment: 48 pages. arXiv admin note: text overlap with arXiv:1011.342
Hilbert's Tenth Problem for function fields of varieties over number fields and p-adic fields
Let k be a subfield of a p-adic field of odd residue characteristic, and let
L be the function field of a variety of dimension n >= 1 over k. Then Hilbert's
Tenth Problem for L is undecidable. In particular, Hilbert's Tenth Problem for
function fields of varieties over number fields of dimension >= 1 is
undecidable.Comment: 19 pages; to appear in Journal of Algebr
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