Rational invariants of a group action. Construction and rewriting

Abstract

Geometric constructions applied to a rational action of an algebraic group lead to a new algorithm for computing rational invariants. A finite generating set of invariants appears as the coefficients of a reduced Gröbner basis. The algorithm comes in two variants. In the first construction the ideal of the graph of the action is considered. In the second one the ideal of a cross-section is added to the ideal of the graph. Zerodimensionality of the resulting ideal brings a computational advantage. In both cases, reduction with respect to the computed Gröbner basis allows us to express any rational invariant in terms of the generators.

Algebraic and Differential Invariants

“…In the second subsection we provide an algorithm to compute a generating set of rational invariants for the rational action of an algebraic group. This is a quick presentation of the main results of [31]. In the last subsection we explain how the algorithm also delivers the normalized invariants as algebraic invariants.…”

“…The coefficients of the Chow form of O z were first shown to form a separating set of rational invariants, and hence a generating set [58]. In [31,44,35] a reduced Gröbner basis for O z is used. Its coefficients are shown to form a generating set of rational invariants by exhibiting an algorithm to rewrite any other rational invariants in terms of those.…”

“…The distinguishing feature of [31] is to additionaly incorporate the idea of the geometric construction reviewed in the previous section. In the present context, a cross-section of degree e is an irreducible subvariety P of Z that intersects the generic orbits in e distinct points.…”

“…This article reviews results of [14,28,29,30,31,32] in order to show their coherence in addressing algorithmically an algebraic description of the differential invariants of a group action. In the first section we show how to compute the rational invariants of a group action and give concrete expressions to a set of algebraic invariants that are of fundamental importance in the differential context.…”

Reduction of Algebraic Parametric Systems by Rectification of Their Affine Expanded Lie Symmetries

“…Thus, we restrict ourself to Lie symmetries associated to affine infinitesimal generators for which invariant coordinates computation is easy (for general case see [5] and references therein). Hence, we do not follows methods developed for general cases because their complexity are likely exponential in input's size while we focus our attention to method of quasi-polynomial complexity.…”