Rational Invariants of Even Ternary Forms Under the Orthogonal Group

Abstract

In this article we determine a generating set of rational invariants of minimal cardinality for the action of the orthogonal group $$\\mathrm {O}_{3}$$O3 on the space $$\\mathbb {R}[x,y,z]_{2d}$$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 $$\\mathrm {B}_{3}$$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 $$\\mathrm {B}_{3}$$B3-equivariance properties. Our explicit construction of these bases should be relevant well beyond the scope of this paper. The expression of the $$\\mathrm {B}_{3}$$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 $$\\mathrm {O}_{3}$$O3-invariants, we provide efficient algorithms for their evaluation and rewriting. We also use the constructed $$\\mathrm {B}_{3}$$B3-invariants to determine the $$\\mathrm {O}_{3}$$O3-orbit locus and provide an algorithm for the inverse problem of finding an element in $$\\mathbb {R}[x,y,z]_{2d}$$R[x,y,z]2d with prescribed values for its invariants. These computational issues are relevant in brain imaging.