Rational invariants of even ternary forms under the orthogonal group

Paul Görlach (1) , Evelyne Hubert (2, 3) , Théo Papadopoulo (3, 4)

1 MPI-MiS - Max-Planck-Institut für Mathematik in den Naturwissenschaften
2 AROMATH - AlgebRe, geOmetrie, Modelisation et AlgoriTHmes
3 COMUE UCA - COMUE Université Côte d'Azur (2015-2019)
4 ATHENA - Computational Imaging of the Central Nervous System

In this article we determine a generating set of rational invariants of minimal cardinality for the action of the orthogonal group O3 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 determining the invariants for the action on a subspace of the finite subgroup 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 B3-equivariance properties. Our explicit construction of these bases should be relevant well beyond the scope of this paper. The expression of the 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 O3-invariants, we provide efficient algorithms for their evaluation and rewriting. We also use the constructed B3-invariants to determine the O3-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.

Domaines: Calcul formel [cs.SC]; Géométrie algébrique [math.AG]

Paul Görlach, Evelyne Hubert, Théo Papadopoulo. Rational invariants of even ternary forms under the orthogonal group. Foundations of Computational Mathematics, 2019, 19, pp.1315-1361. ⟨10.1007/s10208-018-9404-1⟩. ⟨hal-01570853v4⟩