Kato-Milne Cohomology and Polynomial Forms

Given a prime number $p$, a field $F$ with $\operatorname{char}(F)=p$ 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 $\widetilde C_{p,m}$ 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 $\widetilde C_{p,m}$ field $F$, the symbol length of $H_p^2(F)$ is bounded from above by $p^{m-1}-1$ and for any $n \geq \lceil (m-1) \log_2(p) \rceil+1$, $H_p^{n+1}(F)=0$.