#### Conference abstracts

Session B3 - Symbolic Analysis

July 13, 14:30 ~ 14:55 - Room B2

## Invariants of ternary quartics under the orthogonal group

### Inria Méditerranée, France   -   evelyne.hubert@inria.fr

Classical invariant theory has essentially addressed the action of the general linear group on homogeneous polynomials. Yet the orthogonal group arises in applications as the relevant group of transformations, especially in 3 dimensional space. Having a complete set of invariants for its action on quartics is, for instance, relevant in determining biomarkers for white matter from diffusion MRI [1]

We characterize a generating set of rational invariants of the orthogonal group by reducing the problem to the action of a finite subgroup on a slice. The invariants of the orthogonal group can then be obtained in an explicit way. But their numerical evaluation can be achieved more straightforwardly. These results can furthermore be generalised to sextics and higher even degree forms thanks to a novel and explicit construction of a basis of harmonic polynomials at each degree.

[1] A.~Ghosh, T.~Papadopoulo, and R.~Deriche. Biomarkers for hardi: 2nd \& 4th order tensor invariants. In {\em {IEEE International Symposium on Biomedical Imaging: From Nano to Macro - 2012}}, 2012.

Joint work with Paul Görlach (Inria \& Max Planck Institute Leipzig), Théo Papadopoulo (Inria).

FoCM 2017, based on a nodethirtythree design.