Let $F$ denote a binary form of order $d$ over the complex numbers. If $r$ is a divisor of $d$ , then the Hilbert covariant ${{H}_{r,\,d}}\,\left( F \right)$ vanishes exactly when $F$ is the perfect power of an order $r$ form. In geometric terms, the coefficients of $H$ give defining equations for the image variety $X$ of an embedding ${{\text{P}}^{r}}\,\to \,{{\text{P}}^{d}}$ . In this paper we describe a new construction of the Hilbert covariant and simultaneously situate it into a wider class of covariants called the Göttingen covariants, all of which vanish on $X$ . We prove that the ideal generated by the coefficients of $H$ defines $X$ as a scheme. Finally, we exhibit a generalisation of the Göttingen covariants to $n$ -ary forms using the classical Clebsch transfer principle.

Let A, B denote binary forms of order d, and let 2r−1 = (A, B)2r−1 be the sequence of their linear combinants for . It is known that 1, 3 together determine the pencil {A + λ B}λ∈P1 and hence indirectly the higher combinants 2r−1. In this paper we exhibit explicit formulae for all r ≥ 3, which allow us to recover 2r−1 from the knowledge of 1 and 3. The calculations make use of the symbolic method in classical invariant theory, as well as the quantum theory of angular momentum. Our theorem pertains to the plethysm representation ∧2Sd for the group SL2. We give an example for the group SL3 to show that such a result may hold for other categories of representations.

This is a note on the classical Waring's problem for algebraic forms. Fix integers $(n,d,r,s)$ , and let $\Lambda $ be a general $r$ -dimensional subspace of degree $d$ homogeneous polynomials in $n+1$ variables. Let $\mathcal{A}$ denote the variety of $s$ -sided polar polyhedra of $\Lambda $ . We carry out a case-by-case study of the structure of $\mathcal{A}$ for several specific values of $(n,d,r,s)$ . In the first batch of examples, $\mathcal{A}$ is shown to be a rational variety. In the second batch, $\mathcal{A}$ is a finite set of which we calculate the cardinality.