Rational invariants of even ternary forms under the orthogonal group
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In this article 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 one hand, the Slice Lemma allows us to reduce the problem to dermining 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.
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