Rational, Replacement, and Local Invariants of a Group Action

The paper presents a new algorithmic construction of a finite generating set of rational invariants for the rational action of an algebraic group on the affine space. The construction provides an algebraic counterpart of the moving frame method in differential geometry. The generating set of rational invariants appears as the coefficients of a Groebner basis, reduction with respect to which allows to express a rational invariant in terms of the generators. The replacement invariants, introduced in the paper, are tuples of algebraic functions of the rational invariants. Any invariant, whether rational, algebraic or local, can be can be rewritten terms of replacement invariants by a simple substitution.Comment: 37 page

Differential Invariants of Conformal and Projective Surfaces

We show that, for both the conformal and projective groups, all the differential invariants of a generic surface in three-dimensional space can be written as combinations of the invariant derivatives of a single differential invariant. The proof is based on the equivariant method of moving frames.Comment: This is a contribution to the Proceedings of the 2007 Midwest Geometry Conference in honor of Thomas P. Branson, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Rational Invariants of a Group Action

National audienceThis article is based on an introductory lecture delivered at the Journées Nationales de Calcul Formel that took place at the Centre International de Recherche en Mathématiques (2013) in Marseille. We introduce basic notions on algebraic group actions and their invariants. Based on geometric consideration, we present algebraic constructions for a generating set of rational invariants

Differential Algebra for Derivations with Nontrivial Commutation Rules

The classical assumption of differential algebra, differential elimination theory and formal integrability theory is that the derivations do commute. That is the standard case arising from systems of partial differential equations written in terms of the derivations w.r.t. the independant variables. We inspect here the case where the derivations satisfy nontrivial commutation rules. That situation arises for instance when we consider a system of equations on the differential invariants of a Lie group action. We develop the algebraic foundations for such a situation. They lead to algorithms for completion to formal integrability and differential elimination

Rational Invariants of a Group Action

National audienceThis article is based on an introductory lecture delivered at the Journées Nationales de Calcul Formel that took place at the Centre International de Recherche en Mathématiques (2013) in Marseille. We introduce basic notions on algebraic group actions and their invariants. Based on geometric consideration, we present algebraic constructions for a generating set of rational invariants

Invariant Algebraic Sets and Symmetrization of Polynomial Systems

International audienceAssuming the variety of a polynomial set is invariant under a group action, we construct a set of invariants that define the same variety. Our construction can be seen as a generalization of the previously known construction for finite groups. The result though has to be understood outside an invariant variety which is independent of the polynomial set considered. We introduce the symmetrizations of a polynomial that are polynomials in a generating set of rational invariants. The generating set of rational invariants and the symmetrizations are constructed w.r.t. a section to the orbits of the group action

Convolution Surfaces based on Polygons for Infinite and Compact Support Kernels

International audienceWe provide formulae to create 3D smooth shapes fleshing out a skeleton made of line segments and planar polygons. The boundary of the shape is a level set of the convolution function obtained by integration along the skeleton. The convolution function for a complex skeleton is thus the sum of the convolution functions for the basic elements of the skeleton. Providing formulae for the convolutionof a polygon is the main contribution of the present paper. We improve on previous results in several ways. First we do not require the prior triangulation of the polygon. Then, we obtain formulae for families of kernels, either with infinite or compact supports. Last, but not least, in the case of compact support kernels, the geometric computations needed are restricted to intersections of spheres with line segments

Differential invariants of a Lie group action: syzygies on a generating set

Given a group action, known by its infinitesimal generators, we exhibit a complete set of syzygies on a generating set of differential invariants. For that we elaborate on the reinterpretation of Cartan's moving frame by Fels and Olver (1999). This provides constructive tools for exploring algebras of differential invariants.Comment: Journal of Symbolic Computation (2008

Rational invariants of even ternary forms under the orthogonal group

In this article we determine a generating set of rational invariants of minimal cardinality for the action of the orthogonal group $\mathrm{O}_3$ on the space $\mathbb{R}[x,y,z]_{2d}$ of ternary forms of even degree $2d$. The construction relies on two key ingredients: On one hand, the Slice Lemma allows us to reduce the problem to dermining the invariants for the action on a subspace of the finite subgroup $\mathrm{B}_3$ of signed permutations. On the other hand, our construction relies in a fundamental way on specific bases of harmonic polynomials. These bases provide maps with prescribed $\mathrm{B}_3$-equivariance properties. Our explicit construction of these bases should be relevant well beyond the scope of this paper. The expression of the $\mathrm{B}_3$-invariants can then be given in a compact form as the composition of two equivariant maps. Instead of providing (cumbersome) explicit expressions for the $\mathrm{O}_3$-invariants, we provide efficient algorithms for their evaluation and rewriting. We also use the constructed $\mathrm{B}_3$-invariants to determine the $\mathrm{O}_3$-orbit locus and provide an algorithm for the inverse problem of finding an element in $\mathbb{R}[x,y,z]_{2d}$ with prescribed values for its invariants. These are the computational issues relevant in brain imaging.Comment: v3 Changes: Reworked presentation of Neuroimaging application, refinement of Definition 3.1. To appear in "Foundations of Computational Mathematics