Foundations of Computational Mathematics

, Volume 19, Issue 6, pp 1315–1361 | Cite as

Rational Invariants of Even Ternary Forms Under the Orthogonal Group

  • Paul Görlach
  • Evelyne HubertEmail author
  • Théo Papadopoulo


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 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 \(\mathrm {B}_{3}\) 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 computational issues are relevant in brain imaging.


Computational invariant theory Harmonic polynomials Orthogonal group Slice Rational invariants Diffusion MRI Neuroimaging 

Mathematics Subject Classification

12Y05 13A50 13P25 14L24 14Q99 20B30 20C30 33C55 42C05 68U10 68W30 



Paul Görlach was partly funded by INRIA Mediterranée Action Transverse. Evelyne Hubert wishes to thank Rachid Deriche, Frank Grosshans, Boris Kolev for discussions and valuable pointers. Théo Papadopoulo receives funding from the ERC Advanced Grant No. 694665 : CoBCoM—Computational Brain Connectivity Mapping.


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Copyright information

© SFoCM 2018

Authors and Affiliations

  • Paul Görlach
    • 1
    • 2
  • Evelyne Hubert
    • 1
    Email author
  • Théo Papadopoulo
    • 1
  1. 1.INRIA MéditerranéeSophia AntipolisFrance
  2. 2.Max Planck Institute for Mathematics in the SciencesLeipzigGermany

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