Let $Q^{(n, d)}$ be a set of real polynomials in $n$ variables of degree $d$, and $S_k^{(n, d)}$ the set of the sum of squares of any $k$ such polynomials: \begin{equation} S_k^{(n, d)} := \left\lbrace \sum_{i = 1}^k q_i^2\left(x_1, x_2, \ldots, x_n\right) \ \middle| \ q_i \in Q^{(n, d)}, \ i \in \left\lbrace 1, 2, \ldots, k \right\rbrace \right\rbrace. \end{equation} My question is: what is the smallest $k$ such that it holds that (possibly depending on $n$ and $d$) \begin{equation} S_k^{(n, d)} = S_{k + 1}^{(n, d)} = S_{k + 2}^{(n, d)} = \ldots \end{equation} that is, how many polynomial squares at most are needed to represent any sum of any number of polynomial squares?
What I've found so far:
- For $n = 1$ (univariate polynomials), only $2$ squares are necessary to represent any sum of any number of squares of polynomials, regardless of the degree. When $d = 1$, at most $n + 1$ squares are necessary, regardless of the number of variables. When $n = 2$ and $d = 2$, we need at most $3$ squares (all stated in this article).
- Using the square matrical representation we can, in worst case, express some such sum as a sum of $n + d \choose n$ polynomial squares, but the solution is neither guaranteed to be unique, nor optimal in the number of squares.
- Any sum of squares of rational functions in $n$ variables can be expressed as sum of $2^n$ squares of rational functions (stated in the same article, due to Pfister's work in 1967.).
Other connected results I've found are often about proving whether a polynomial is positive semidefinite, but this is a broader class of polynomials than the one I am interested in. I was hoping that maybe $k$ is at least $\le 2^n$, that is, that Pfister's result would still hold if we replaced rational functions with simple polynomials; but I am unable to prove this on my own.