I’m looking at two different models for learning polynomial functions, and trying to determine if they are equivalent. After a couple days of thinking, I’ve reduced the question to the following:

Can every symmetric polynomial of degree \(r\) in \(d\) variables that has no constant term be written as a sum of the \(r\)-th powers of linear polynomials in \(d\) degrees and a homogeneous polynomial of degree \(r\) each of whose monomials involves at most \(d-1\) variables?