Zero subspaces of polynomials on ℓ1(Γ)

We provide two examples of complex homogeneous quadratic polynomials P on Banach spaces of the form ℓ1(Γ). The first polynomial P has both separable and nonseparable maximal zero subspaces. The second polynomial P has the property that while the index-set Γ is not countable, all zero subspaces of P are separable.