Rational and replacement invariants of a group action

Group actions are ubiquitous in mathematics. They arise indiverse areas of applications, from classical mechanics to computervision. A classical but central problem is to compute a generatingset of invariants. We consider a rational group action on the affine space andpropose a construction of a finite set of rational invariants and asimple algorithm to rewrite any rational invariant in terms ofthose generators. The construction comes into two variants bothconsisting in computing a reduced Gröbner basis of apolynomial ideal. That polynomial ideal is of dimension zero in thesecond variant that relies on the choice of a cross-section, avariety that intersects generic orbits in a finite number ofpoints. A generic linear space of complementary dimension to theorbits can be chosen for cross-section. When the intersection of a generic orbit with the cross-sectionconsists of a single point, the rewriting of any rational invariantin terms of the computed generating set trivializes into areplacement. For general cross-sections we introduce a finite setof replacement invariants that are algebraicfunctions of the rational invariants. Any rational invariant can berewritten in terms of those by simple substitution. We have therefore obtained an algebraic formulation of themoving frame construction of Fels and Olver [2], providing a bridgebetween the algebraic theory of polynomial and rational invariants[7, 1], and the differential-geometric theory of local smoothinvariants, [6]. In this abstract we formalize our main results in the case whereK is an algebraically closed field of characteristic zero. Severalexamples, both classical and original, are treated in theposter. Take an algebraic group G given by an unmixeddimensional ideal G in a polynomial ringK[λ1,...,λℓ].A rational action of the group G onKⁿ isgiven as the rational map: [EQUATION] whereh,g1,...,gnare polynomial functions inK[λ1,...,λℓ,z1,...,zn]. An element p/q ∈K(z) is a rationalinvariant ifp(λ·z)q(z)=p(z)q(λz)mod G. The set of rational invariants forms a(finitely generated) fieldK(z)^G. A cross-section of degree d > 0 is anirreducible affine variety P that intersectsgeneric orbits in exactly d simple points. Whengeneric orbits are of dimension r > 0, ageneric linear affine space of codimension r isa cross-section. Consider a new set of variables Z =(Z1,...,Zn).The ideal (Z - λ ·z) is the saturation by h ofthe ideal generated by the polynomialsh(λ,z)Zi-gi(λ,z),1 ≤ i ≤ n. Wedefine the following two elimination ideals: [EQUATION] where P ⊂ K[Z] isthe ideal of the chosen cross-section P. Thevariety of O inKⁿ xKⁿ is the closure of thegraph of the action. If we consider a projection on the secondcomponent Kⁿ, the fiberabove P is the variety ofI. The extensionsO^e andI^e ofthe ideals O and I toK(z)[Z] are bothequidimensional of respective dimension r and 0.They are the heart of our construction. The following results arevalid for any term order chosen on Z.