## Algebra: it matters

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?