Kato–Milne cohomology and polynomial forms

Given a prime number p, a field F with char(F)=p and a positive integer n, we study the class-preserving modifications of Kato–Milne classes of decomposable differential forms. These modifications demonstrate a natural connection between differential forms and p-regular forms. A p-regular form is defined to be a homogeneous polynomial form of degree p for which there is no nonzero point where all the order p−1 partial derivatives vanish simultaneously. We define a C˜_p,m field to be a field over which every p-regular form of dimension greater than p^m is isotropic. The main results are that for a C˜_p,m field F, the symbol length of H_p ²(F) is bounded from above by p^m−1−1 and for any n⩾⌈(m−1)log₂⁡(p)⌉+1, H_p ⁿ⁺¹(F)=0.