Sums of squares of linear forms

The length and other invariants of a real field

“…In [6] it is shown that, if u(K (i)) ≤ 2 n with n ≥ 2, then u(K ) ≤ 2 n+2 −2n −6, improving for n ≥ 3 a bound from [9]. The following improvement of a similar bound in [2] on the length was pointed out to us by C. Scheiderer.…”

“…So ϕ contains a subform a, a with a ∈ K × . Now, a ∈ D K (ϕ) ⊆ K 2 = D K (2) yields that a, a = 1, 1 . This shows that every (m +1)-dimensional totally positive form over K contains 1, 1 , so any m-dimensional totally positive form over K represents 1.…”

“…Let k be a real field with p(k) ≥ 3 and fix a ∈ D k (3)\D k (2). Consider the class of real field extensions K /k such that a / ∈ D K (2).…”

“…Then ϕ K (i) is isotropic, so K (i)(ϕ) is a rational function field over K (i), and this allows to conclude that the extended form (ψ ⊥ −1 ) K (i)(ϕ) is still essential. Since ϕ is totally indefinite over K it must be isotropic, so s ∈ D K (2). Therefore p(K ) ≤ 2.…”

“…One of the aims of this article is to investigate the relations between these invariants for real fields with a special focus on the length. This invariant was introduced in [2] and studied together with a certain function g K : N −→ N ∪ {∞} related to sums of squares of linear forms over K . In Sect.…”