In this carefully written paper the authors continue previous work on positive semidefinite forms (say of degree \(m\)) \(f= f(x_ 1, \dots, x_ n)\in \mathbb{R}[x_ 1, \dots, x_ n]= A_ n\). If \(f= \sum_ 1^ t h^ 2_ i\) in \(A_ n\) they associate to this specific expression of \(f\) as a sum of squares a \(t\)-dimensional “Gram matrix” \(V\in M_ t (\mathbb{R})\) and a quadratic form \(Q\) over \(\mathbb{R}\). The minimal possible number \(t\) is called the length of \(f\). They get: \[ \text{length} (f)= \min\{ \text{rank } V\}= \min\{ \text{length } Q\}. \] Via this result, one is able to reduce the notion of length for arbitrary forms of even degree \(m\) to the notion of length of quadratic forms and to problems in linear algebra.
Let \(F_{n,m}\) denote the \(\mathbb{R}\)-space of \(n\)-ary \(m\)-ics in \(A_ n\), and let \(P(n,m)= \sup\{ \text{length} (f)\mid f\in F_{n,m}\), \(f\) is a sum of squares in \(A_ n\}\) be the “Pythagoras number” of \(F_{n,m}\). The main result of the paper contains upper and lower estimates for \(P(n,m)\). In particular it follows that for fixed \(m\), \(P(n,m)\) has the order of magnitude \(n^{m/2}\) for \(n\to\infty\).
The proof uses the geometric-combinatorial method of “cages”. For fixed \(n\), \(m\) a case \(C\) is, roughly speaking, a convex collection of \(n\)-ary \(m\)-ic monomials. The Main Theorem is proved for an arbitrary cage \(C\), not only for the “full” cage \(C_{n,m}\) corresponding to \(F_{n,m}\).
For the entire collection see [Zbl 0812.00023].