19 research outputs found
Integrality at a prime for global fields and the perfect closure of global fields of characteristic p>2
Let k be a global field and \pp any nonarchimedean prime of k. We give a new
and uniform proof of the well known fact that the set of all elements of k
which are integral at \pp is diophantine over k. Let k^{perf} be the perfect
closure of a global field of characteristic p>2. We also prove that the set of
all elements of k^{perf} which are integral at some prime \qq of k^{perf} is
diophantine over k^{perf}, and this is the first such result for a field which
is not finitely generated over its constant field. This is related to Hilbert's
Tenth Problem because for global fields k of positive characteristic, giving a
diophantine definition of the set of elements that are integral at a prime is
one of two steps needed to prove that Hilbert's Tenth Problem for k is
undecidable.Comment: 10 pages; minor revisions mad
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
Undecidability in function fields of positive characteristic
We prove that the first-order theory of any function field K of
characteristic p>2 is undecidable in the language of rings without parameters.
When K is a function field in one variable whose constant field is algebraic
over a finite field, we can also prove undecidability in characteristic 2. The
proof uses a result by Moret-Bailly about ranks of elliptic curves over
function fields.Comment: 12 pages; strengthened main theorem, proved undecidability in the
language of rings without parameter