## Section: New Results

### Rational invariants of even ternary forms under the orthogonal group

Participant : Evelyne Hubert.

In [8], we determine a generating set of rational invariants of minimal cardinality for the action of the orthogonal group ${O}_{3}$ on the space $R{[x,y,z]}_{2d}$ of ternary forms of even degree $2d$. The construction relies on two key ingredients: On the 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 ${B}_{3}$ 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 ${B}_{3}$-equivariance properties. Our explicit construction of these bases should be relevant well beyond the scope of this paper. The expression of the ${B}_{3}$-invariants can then be given in a compact form as the composition of two equivariant maps. Instead of providing (cumbersome) explicit expressions for the ${O}_{3}$-invariants, we provide efficient algorithms for their evaluation and rewriting. We also use the constructed ${B}_{3}$-invariants to determine the ${O}_{3}$-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.

This is a joint work with P. Görlach (Max Planck institute, Leipzig) and T. Papadopoulo (EPI Athena, Inria SAM)