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.

中文翻译：

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正交群下偶三元形式的有理不变量

在本文中，我们确定正交基团\（\ mathrm {O} _ {3} \）在空间\（\ mathbb {R} [x，y，z ]上的作用的最小基数的有理不变量的生成集] _ {2d} \）偶数阶2 d的三元形式。构造依赖于两个关键要素：一方面，切片引理使我们可以减少问题，从而确定对有限子组\（\ mathrm {B} _ {3} \）的子空间上的操作产生不变量有符号排列。另一方面，我们的构造从根本上依赖于谐波多项式的特定基础。这些基础为地图提供了规定的\（\ mathrm {B} _ {3} \）-等方差属性。我们对这些基础的明确构建应该远远超出本文的范围。所述的表达\（\ mathrm {B} _ {3} \） -invariants然后可以在紧凑的形式为两个等变映射的组合物进行说明。我们没有为\（\ mathrm {O} _ {3} \）不变量提供（麻烦的）显式表达式，而是为它们的评估和重写提供了有效的算法。我们还使用构造的\（\ mathrm {B} _ {3} \）-不变式来确定\（\ mathrm {O} _ {3} \）-轨道轨迹，并提供一种算法来解决寻找\（\ mathbb {R} [x，y，z] _ {2d} \中的元素具有其不变式的规定值。这些计算问题与大脑成像有关。