5,239 research outputs found
Geometric Axioms for Differentially Closed Fields with Several Commuting Derivations
A geometric first-order axiomatization of differentially closed fields of
characteristic zero with several commuting derivations, in the spirit of
Pierce-Pillay, is formulated in terms of a relative notion of prolongation for
Kolchin-closed sets
Algorithms yield upper bounds in differential algebra
Consider an algorithm computing in a differential field with several
commuting derivations such that the only operations it performs with the
elements of the field are arithmetic operations, differentiation, and zero
testing. We show that, if the algorithm is guaranteed to terminate on every
input, then there is a computable upper bound for the size of the output of the
algorithm in terms of the input. We also generalize this to algorithms working
with models of good enough theories (including for example, difference fields).
We then apply this to differential algebraic geometry to show that there
exists a computable uniform upper bound for the number of components of any
variety defined by a system of polynomial PDEs. We then use this bound to show
the existence of a computable uniform upper bound for the elimination problem
in systems of polynomial PDEs with delays
Computation of the unipotent radical of the differential Galois group for a parameterized second-order linear differential equation
We propose a new method to compute the unipotent radical of the
differential Galois group associated to a parameterized second-order
homogeneous linear differential equation of the form
where is a rational
function in with coefficients in a -field of characteristic zero,
and is a commuting set of parametric derivations. The procedure developed
by Dreyfus reduces the computation of to solving a creative
telescoping problem, whose effective solution requires the assumption that the
maximal reductive quotient is a -constant linear differential
algebraic group. When this condition is not satisfied, we compute a new set of
parametric derivations such that the associated differential Galois
group has the property that is -constant, and such
that is defined by the same differential equations as . Thus
the computation of is reduced to the effective computation of
. We expect that an elaboration of this method will be successful in
extending the applicability of some recent algorithms developed by Minchenko,
Ovchinnikov, and Singer to compute unipotent radicals for higher order
equations.Comment: 12 page
Planar 2-homogeneous commutative rational vector fields
In this paper we prove the following result: if two 2-dimensional
2-homogeneous rational vector fields commute, then either both vector fields
can be explicitly integrated to produce rational flows with orbits being lines
through the origin, or both flows can be explicitly integrated in terms of
algebraic functions. In the latter case, orbits of each flow are given in terms
of -homogeneous rational functions as curves . An
exhaustive method to construct such commuting algebraic flows is presented. The
degree of the so-obtained algebraic functions in two variables can be
arbitrarily high.Comment: 23 page
Generic derivations on o-minimal structures
Let be a complete, model complete o-minimal theory extending the theory
RCF of real closed ordered fields in some appropriate language . We study
derivations on models . We introduce the notion
of a -derivation: a derivation which is compatible with the
-definable -functions on . We show
that the theory of -models with a -derivation has a model completion
. The derivation in models
behaves "generically," it is wildly discontinuous and its kernel is a dense
elementary -substructure of . If RCF, then
is the theory of closed ordered differential fields (CODF) as introduced by
Michael Singer. We are able to recover many of the known facts about CODF in
our setting. Among other things, we show that has as its open
core, that is distal, and that eliminates
imaginaries. We also show that the theory of -models with finitely many
commuting -derivations has a model completion.Comment: 29 page
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