Binary decompositions and varieties of sums of binaries

Abstract

A homogeneous element in C [ x 0 , … , x n ] 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 = 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 , 8 .