Given a system of regular quadratic forms \(\{q_i : i\in I\}\) with the same underlying vector space \(V\) over \(K\), we say that the system is hyperbolic if \(V\) decomposes as \(W\oplus W\) where each of the quadratic forms vanishes over each copy of \(W\). This of course generalizes the standard notion of a hyperbolic quadratic form, which for a single form means that \(q \simeq x_1y_1+\dots+x_n y_n\).
Suppose \(\operatorname{char}(K)\neq 2\). Then every quadratic form is diagonalizable. For any ordering \(P\) of \(K\) and \(\alpha\in K\), \(\operatorname{sgn}_P(\alpha)\) is 1 when \(\alpha \in P\), \(-1\) when \(\alpha \in -P\) and \(0\) when \(\alpha=0\). Then for \(q=\langle \alpha_1,\dots,\alpha_n \rangle\), \(\operatorname{sgn}_P(q)=\sum_{i=1}^n \operatorname{sgn}_P(\alpha_i)\). The total signature of \(q\), denoted \(\operatorname{sgn}(q)\), is the function mapping each ordering \(P\) of \(K\) to \(\operatorname{sgn}_P(q)\). Pfister proved that there exists a natural number \(m\) for which \(m\times q\) is hyperbolic if and only if \(\operatorname{sgn}(q)=0\).
The goal of the paper under discussion is to extend Pfister’s condition to systems of quadratic forms. The author associates a single quadratic form \(q_{A,\sigma}\) to the system \(\{q_i : i\in I\}\) (\(A\) is the \(K\)-algebra of adjoints of \(\{q_i : i\in I\}\), \(\sigma\) its canonical involution, and \(q_{A,\sigma}\) is the involution-trace quadratic form on \(A\)). The main result (Theorem A) is that there exists a natural number \(m\) for which \(m \times \{q_i : i\in I\}\) if and only if \(\operatorname{sgn}(q_{A,\sigma})=0\). The author presents other interesting results too, some of which require additional assumptions on the base-field or the size of the system.