Relations between invariants of formally real fields (Q1364607)

Let \(F\) be a formally real field. Continuing some of his previous work the author studies the invariant \[ l(F)=\inf \{n\in \mathbb{N}\mid\text{ If }q:F^n\to F\text{ is any totally positive quadratic form then }D(q)=\Sigma(F)\}. \] He introduces another invariant \(c(F)= \inf \{m\in \mathbb{N}\mid\). If \(\omega\in \Sigma(F)\) and \(q:F^{m+1}\to F\) is a totally positive quadratic form then \(q\cong \langle \;\rangle \langle \langle \omega\rangle\rangle\perp q_1\) for some \(a\in F\) and some quadratic form \(q_1\}\). In the first part of the paper he describes various relations of these invariants to the invariants \(ud(F)\), \(\beta_F(i)\) and \(C(F)\). As a main result he proves the equivalence between the finiteness of the invariants \(\beta_F(i)\), \(c(F)\), \(\ell(F)\), \(ud(F)\) and \(\widetilde{ud}(F)\) (Theorem 2.9). In the second part he studies the invariants \(\beta_F(i)\) and \(c(F)\). Under the assumption that \(l(F)< \infty\) and that any totally positive 2-fold Pfister form represents all elements of \(D_F(\langle 1,\alpha\rangle)\) for some \(\alpha\in \Sigma(F)\) he describes the values of \(c(F)\) by means of \(l(F)\) and \(\beta_F(i)\) (Theorem 2.6) and finally gives some estimates for \(c(F)\) (Theorem 3.7).