Rational invariants of even ternary forms under the orthogonal group - Inria - Institut national de recherche en sciences et technologies du numérique Accéder directement au contenu
Article Dans Une Revue Foundations of Computational Mathematics Année : 2019

Rational invariants of even ternary forms under the orthogonal group

Résumé

In this article we determine a generating set of rational invariants of minimal cardinality for the action of the orthogonal group O3 on the space $R[x, y, z]_2d$ 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 determining the invariants for the action on a subspace of the finite subgroup B3 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 B3-equivariance properties. Our explicit construction of these bases should be relevant well beyond the scope of this paper. The expression of the B3-invariants can then be given in a compact form as the composition of two equivariant maps. Instead of providing (cumbersome) explicit expressions for the O3-invariants, we provide efficient algorithms for their evaluation and rewriting. We also use the constructed B3-invariants to determine the O3-orbit locus and provide an algorithm for the inverse problem of finding an element in $R[x, y, z]_2d$ with prescribed values for its invariants. These are the computational issues relevant in brain imaging.
Fichier principal
Vignette du fichier
GoHuPa17.pdf (2.12 Mo) Télécharger le fichier
Origine Fichiers produits par l'(les) auteur(s)
Loading...

Dates et versions

hal-01570853 , version 1 (31-07-2017)
hal-01570853 , version 2 (18-08-2017)
hal-01570853 , version 3 (08-07-2018)
hal-01570853 , version 4 (12-07-2018)

Identifiants

Citer

Paul Görlach, Evelyne Hubert, Théo Papadopoulo. Rational invariants of even ternary forms under the orthogonal group. Foundations of Computational Mathematics, 2019, 19, pp.1315-1361. ⟨10.1007/s10208-018-9404-1⟩. ⟨hal-01570853v4⟩
562 Consultations
354 Téléchargements

Altmetric

Partager

Gmail Mastodon Facebook X LinkedIn More