ON THE h-INVARIANT OF CUBIC FORMS, AND SYSTEMS OF CUBIC FORMS

We define a so-called `-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 `-invariant of a system of rational forms F1, . . . , Fr in terms of the `-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 400000r4 variables has a non-trivial rational zero.