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Abstract: We study the following question in this paper: If p is a prime, m a positive integer, and an arbitrary sequence consisting of “Y” or “N,” does there exist a division algebra of exponent over a valued field such that the underlying division algebra of the tensor power has a valuation extending v if and only if ? We show that if such an algebra exists, then its index must be bounded below by a power of p that depends on both m and S, and we then answer the question affirmatively by constructing such an algebra of minimal index. [Copyright &y& Elsevier]

Abstract: In an earlier paper [T. Košir, B.A. Sethuraman, Determinantal varieties over truncated polynomial rings, J. Pure Appl. Algebra 195 (2005) 75–95] we had begun a study of the components and dimensions of the spaces of th order jets of the classical determinantal varieties: these are the varieties obtained by considering generic () matrices over rings of the form , and for some fixed r, setting the coefficients of powers of t of all minors to zero. In this paper, we consider the case where , and provide a Groebner basis for the ideal which defines the tangent bundle to the classical determinantal variety. We use the results of these Groebner basis calculations to describe the components of the varieties where r is arbitrary. (The components of and were already described in the above cited paper.) [Copyright &y& Elsevier]