A Waring decomposition of a (homogeneous) polynomial f is a minimal sum of powers of linear forms expressing f. Under certain conditions, such a decomposition is unique. We discuss some algorithms to compute the Waring decomposition, which are linked to the equation of certain secant varieties and to eigenvectors of tensors. In particular we explicitly decompose a general cubic polynomial in three variables as the sum of five cubes (Sylvester Pentahedral Theorem).Comment: 32 pages; three Macaulay2 files as ancillary files. Revised with referee's suggestions. Accepted JS

Abstract. Techniques from representation theory, symbolic computational algebra, and numerical algebraic geometry are used to find the minimal generators of the ideal of the trifocal variety. An effective test for determining whether a given tensor is a trifocal tensor is also given.

The variety of principal minors of n × n symmetric matrices, denoted Z n , is invariant under the action of a group G ⊂ GL(2 n ) isomorphic to SL (2) ×n S n . We describe an irreducible G-module of degree-four polynomials constructed from Cayley's 2 × 2 × 2 hyperdeterminant and show that it cuts out Z n settheoretically. This solves the set-theoretic version of a conjecture of Holtz and Sturmfels. Standard techniques from representation theory and geometry are explored and developed for the proof of the conjecture and may be of use for studying similar G-varieties.