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    Essential dimension of simple algebras with involutions

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    Let 1mn1\leq m \leq n be integers with mnm|n and \cat{Alg}_{n,m} the class of central simple algebras of degree nn and exponent dividing mm. In this paper, we find new, improved upper bounds for the essential dimension and 2-dimension of \cat{Alg}_{n,2}. In particular, we show that \ed_{2}(\cat{Alg}_{16,2})=24 over a field FF of characteristic different from 2.Comment: Sections 1 and 3 are rewritte

    Essential dimension of simple algebras in positive characteristic

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    Let pp be a prime integer, 1sr1\leq s\leq r integers, FF a field of characteristic pp. Let \cat{Dec}_{p^r} denote the class of the tensor product of rr pp-symbols and \cat{Alg}_{p^r,p^s} denote the class of central simple algebras of degree prp^r and exponent dividing psp^s. For any integers s<rs<r, we find a lower bound for the essential pp-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\ch(F)=p and ch(F)=2\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.Comment: Any comments are welcom
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