Essential dimension of simple algebras in positive characteristic

Let $p$ be a prime integer, $1\leq s\leq r$ integers, $F$ a field of characteristic $p$. Let $\cat{Dec}_{p^r}$ denote the class of the tensor product of $r$ $p$-symbols and $\cat{Alg}_{p^r,p^s}$ denote the class of central simple algebras of degree $p^r$ and exponent dividing $p^s$. For any integers $s<r$, we find a lower bound for the essential $p$-dimension of $\cat{Alg}_{p^r,p^s}$. Furthermore, we compute upper bounds for $\cat{Dec}_{p^r}$ and $\cat{Alg}_{8,2}$ over $\ch(F)=p$ and $\ch(F)=2$, respectively. As a result, we show $\ed_{2}(\cat{Alg}_{4,2})=\ed(\cat{Alg}_{4,2})=\ed_{2}(\gGL_{4}/\gmu_{2})=\ed(\gGL_{4}/\gmu_{2})=3$ and $3\leq \ed(\cat{Alg}_{8,2})=\ed(\gGL_{8}/\gmu_{2})\leq 10$ over a field of characteristic 2.

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