532 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
The Diophantine problem in Chevalley groups
In this paper we study the Diophantine problem in Chevalley groups , where is an indecomposable root system of rank , is
an arbitrary commutative ring with . We establish a variant of double
centralizer theorem for elementary unipotents . This theorem is
valid for arbitrary commutative rings with . The result is principle to show
that any one-parametric subgroup , , is Diophantine
in . Then we prove that the Diophantine problem in is
polynomial time equivalent (more precisely, Karp equivalent) to the Diophantine
problem in . This fact gives rise to a number of model-theoretic corollaries
for specific types of rings.Comment: 44 page
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