Binary decompositions and varieties of sums of binaries

A homogeneous element in C[x0,…,xn]C[x0,…,xn] is a binary form if it “essentially” involves two variables. The problem of decomposing a generic form as a sum of binary forms is introduced and studied. In the n=2n=2 case a complete answer to a Waring type problem is given and the space parameterizing all the binary decompositions of a generic form is studied: its dimension and its degree are determined and a more detailed description is given for forms of degree d=2,3,5,8d=2,3,5,8.