Denote by $W$ the space of all symmetric bilinear forms on $\mathbb{R}^n$.

Let $Q$ be a quadratic form on $W$. Suppose that $Q(b)\geqslant 0$ for any $b\in W$ such that $b(X,Y)=\ell(X)\cdot\ell(Y)$ for some linear function $\ell$.

Is it possible to find a nonnegative quadratic form $\tilde Q$ such that $\tilde Q(b)=Q(b)$ for any $b$ as above?

Comments

• The forms $b$ (up to sign) can be described as solutions of a system of certain quadratic equations on $W$. This leads to a more general version of this question, but it has a negative answer (thanks to Terry Tao).

• I know that the answer is yes for $n=2$.

Denote by $W$ the space of all symmetric bilinear forms on $\mathbb{R}^n$.

Let $Q$ be a quadratic form on $W$. Suppose that $Q(b)\geqslant 0$ for any $b\in W$ such that $b(X,Y)=\ell(X)\cdot\ell(Y)$ for some linear function $\ell$.

Is it possible to find a nonnegative quadratic form $\tilde Q$ such that $\tilde Q(b)=Q(b)$ for any $b$ as above?

Comments

• The forms $b$ as above (up to sign) can be described by quadratic equations $$b(X,Y)\cdot b(Z,W)=b(X,Z)\cdot b(Y,W).$$ This leads to a more general version of this question, but it has a negative answer (thanks to Terry Tao).

• I know that the answer is yes for $n=2$.