Rational and replacement invariants of a group action
Authors: Hubert Evelyne , Kogan Irina 
Published: January 2007

Group actions are ubiquitous in mathematics. They arise in
diverse areas of applications, from classical mechanics to computer
vision. A classical but central problem is to compute a generating
set of invariants.

We consider a rational group action on the affine space and
propose a construction of a finite set of rational invariants and a
simple algorithm to rewrite any rational invariant in terms of
those generators. The construction comes into two variants both
consisting in computing a reduced Gr&ouml;bner basis of a
polynomial ideal. That polynomial ideal is of dimension zero in the
second variant that relies on the choice of a cross-section, a
variety that intersects generic orbits in a finite number of
points. A generic linear space of complementary dimension to the
orbits can be chosen for cross-section.

When the intersection of a generic orbit with the cross-section
consists of a single point, the rewriting of any rational invariant
in terms of the computed generating set trivializes into a
replacement. For general cross-sections we introduce a finite set
of <i>replacement invariants</i> that are algebraic
functions of the rational invariants. Any rational invariant can be
rewritten in terms of those by simple substitution.

We have therefore obtained an algebraic formulation of the
moving frame construction of Fels and Olver [2], providing a bridge
between the algebraic theory of polynomial and rational invariants
[7, 1], and the differential-geometric theory of local smooth
invariants, [6].

In this abstract we formalize our main results in the case where
K is an algebraically closed field of characteristic zero. Several
examples, both classical and original, are treated in the
poster.

Take an algebraic group <i>G</i> given by an unmixed
dimensional ideal <i>G</i> in a polynomial ring
K[&lambda;<inf>1</inf>,...,&lambda;<inf>&ell;</inf>].
A rational action of the group <i>G</i> on
<i>K</i><sup><i>n</i></sup> is
given as the rational map:

[EQUATION]

where
<i>h,g</i>1,...,<i>g</i><inf><i>n</i></inf>
are polynomial functions in
K[&lambda;<inf>1</inf>,...,&lambda;<inf>&ell;</inf>,<i>z</i><inf>1</inf>,...,<i>z</i><inf><i>n</i></inf>].

An element <i>p</i>/<i>q</i> &isin;
<i>K</i>(<i>z</i>) is a <i>rational
invariant</i> if
<i>p</i>(&lambda;&middot;<i>z</i>)<i>q</i>(<i>z</i>)
=
<i>p</i>(<i>z</i>)<i>q</i>(&lambda;<i>z</i>)
mod <i>G</i>. The set of rational invariants forms a
(finitely generated) field
K(<i>z</i>)<sup><i>G</i></sup>.

A <i>cross-section of degree d</i> &gt; 0 is an
irreducible affine variety <i>P</i> that intersects
generic orbits in exactly <i>d</i> simple points. When
generic orbits are of dimension <i>r</i> &gt; 0, a
generic linear affine space of codimension <i>r</i> is
a cross-section.

Consider a new set of variables <i>Z</i> =
(<i>Z</i><inf>1</inf>,...,<i>Z</i><inf><i>n</i></inf>).
The ideal (<i>Z</i> - &lambda; &middot;
<i>z</i>) is the saturation by <i>h</i> of
the ideal generated by the polynomials
<i>h</i>(&lambda;,<i>z</i>)<i>Z</i><inf><i>i</i></inf>
-
<i>g</i><inf><i>i</i></inf>(&lambda;,<i>z</i>),
1 &le; <i>i</i> &le; <i>n</i>. We
define the following two elimination ideals:

[EQUATION]

where <i>P</i> &sub; K[<i>Z</i>] is
the ideal of the chosen cross-section <i>P</i>. The
variety of <i>O</i> in
K<sup><i>n</i></sup> x
K<sup><i>n</i></sup> is the closure of the
graph of the action. If we consider a projection on the second
component K<sup><i>n</i></sup>, the fiber
above <i>P</i> is the variety of
<i>I</i>.

The extensions
<i>O</i><sup><i>e</i></sup> and
<i>I</i><sup><i>e</i></sup> of
the ideals <i>O</i> and <i>I</i> to
K(<i>z</i>)[<i>Z</i>] are both
equidimensional of respective dimension <i>r</i> and 0.
They are the heart of our construction. The following results are
valid for any term order chosen on <i>Z.</i>