FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2008, VOLUME 14, NUMBER 4, PAGES 167-180

**Dimension polynomials of intermediate differential fields and the
strength of a system of differential equations with group action**

A. B. Levin

Abstract

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Let $K$ be
a differential field of zero characteristic with a basic set
of derivations $$D =
{d_{1},...,d_{m}}
and let $$Q denote the free
commutative semigroup of all elements of the form $$q = d_{1}^{k1}...d_{m}^{km}
where $k$_{i} Î
**N** ($1$£ i £
m).
Let the order of such an element be defined as $ord$q = å_{i=1}^{m}
k_{i}, and for
any $r$Î **N**, let
$$Q(r) =
{q Î Q |
ord q
£r}.
Let $L\; =\; K$áh_{1},...,h_{s}ñ be
a differential field extension of $K$ generated by
a finite set $$h = {h_{1},...,h_{s}}
and let $F$ be an intermediate
differential field of the extension $L/K$.
Furthermore, for any $r$Î **N**, let
$L$_{r} =
K(È_{i=1}^{s}
Q(r)
h_{i})
and $F$_{r} = L_{r}
Ç F.
We prove the existence and describe some properties of
a polynomial $$j_{K,F,h}(t) Î
**Q**[t] such that $$j_{K,F,h}(r) = trdeg_{K}F_{r}
for all sufficiently large $r$Î **N**.
This result implies the existence of a dimension polynomial that
describes the strength of a system of differential equations with
group action in the sense of A. Einstein.
We shall also present a more general result, a theorem on
a multivariate dimension polynomial associated with an
intermediate differential field $F$ and partitions of the
basic set $$D.

Location: http://mech.math.msu.su/~fpm/eng/k08/k084/k08410h.htm

Last modified: February 28, 2009