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 pm is isotropic. The main results are that for a C˜p,m field F, the symbol length of Hp2(F) is bounded from above by pm−1−1 and for any n⩾⌈(m−1)log2⁡(p)⌉+1, Hpn+1(F)=0.