On the h-invariant of Cubic Forms, and Systems of Cubic Forms

We define a so-called ℓ-invariant for systems of homogeneous forms of the same degree, which coincides with the well-known h-invariant for a single quadratic or cubic form, and bound the ℓ-invariant of a system of rational forms F1,…,Fr in terms of the ℓ-invariant of a single form α1F1+…+αrFr in their complex pencil in case of algebraic α1,…,αr⁠. As an application, we show that a system of r rational cubic forms in more than 400 000r4 variables has a non-trivial rational zero.