112 research outputs found
Independence in computable algebra
We give a sufficient condition for an algebraic structure to have a
computable presentation with a computable basis and a computable presentation
with no computable basis. We apply the condition to differentially closed, real
closed, and difference closed fields with the relevant notions of independence.
To cover these classes of structures we introduce a new technique of safe
extensions that was not necessary for the previously known results of this
kind. We will then apply our techniques to derive new corollaries on the number
of computable presentations of these structures. The condition also implies
classical and new results on vector spaces, algebraically closed fields,
torsion-free abelian groups and Archimedean ordered abelian groups.Comment: 24 page
Scott Ranks of Classifications of the Admissibility Equivalence Relation
Let be a recursive language. Let be the set of
-structures with domain . Let be a function with the property that
for all , if and only if
. Then there is some
so that
Differential-algebraic jet spaces preserve internality to the constants
This paper concerns the model theory of jet spaces (i.e., higher-order
tangent spaces) in differentially closed fields. Suppose p is the generic type
of the jet space to a finite dimensional differential-algebraic variety at a
generic point. It is shown that p satisfies a certain strengthening of almost
internality to the constant field called "preserving internality to the
constants". This strengthening is a model-theoretic abstraction of the generic
behaviour of jet spaces in complex-analytic geometry. A counterexample is
constructed showing that only this generic analogue holds in
differential-algebraic geometry.Comment: 13 page
Nonstandard methods for bounds in differential polynomial rings
Motivated by the problem of the existence of bounds on degrees and orders in
checking primality of radical (partial) differential ideals, the nonstandard
methods of van den Dries and Schmidt ["Bounds in the theory of polynomial rings
over fields. A nonstandard approach.", Inventionnes Mathematicae, 76:77--91,
1984] are here extended to differential polynomial rings over differential
fields. Among the standard consequences of this work are: a partial answer to
the primality problem, the equivalence of this problem with several others
related to the Ritt problem, and the existence of bounds for characteristic
sets of minimal prime differential ideals and for the differential
Nullstellensatz.Comment: 18 page
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