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>