The ring of projective invariants of $n$ ordered points on the projective line is one of the most basic and earliest studied examples in Geometric Invariant Theory. It is a remarkable fact and the point of this paper that, unlike its close relative the ring of invariants of $n$ unordered points, this ring can be completely and simply described. In 1894 Kempe found generators for this ring, thereby proving the First Main Theorem for it (in the terminology introduced by Weyl). In this paper we compute...

Let F be a homogeneous polynomial of degree d in m + 1 variables defined over an algebraically closed field of characteristic 0 and suppose that F belongs to the sth secant variety of the d-uple Veronese embedding of ${\mathbb{P}}^m$ into ${\mathbb{P}}^{\genfrac{}{}{0pt}{}{m+d}{d}-1}$ but that its minimal decomposition as a sum of dth powers of linear forms requires more than s summands. We show that if s ≤ d then F can be uniquely written as $F=M{₁}^d+\cdots +M_t^d+Q$, where $M₁,...,M_t$ are linear forms with t ≤ (d-1)/2, and Q is a binary form such that $Q={\sum }_{i=1}^q{l}_i^{d-d_i}{m}_i$ with $l_i$’s linear forms and $m_i$’s forms...