Abstract. Let p be a multilinear polynomial in several non-commuting variables with coefficients in a quadratically closed field K of any characteristic. It has been conjectured that for any n, the image of p evaluated on the set Mn(K) of n by n matrices is either zero, or the set of scalar matrices, or the set sln(K) of matrices of trace 0, or all of Mn(K). We prove the conjecture for n = 2, and show that although the analogous assertion fails for completely homogeneous polynomials, one can salvage the conjecture in this case by including the set of all non-nilpotent matrices of trace zero and also permitting dense subsets of Mn(K).
Let p be a polynomial in several non-commuting variables with coefficients in a field K of arbitrary characteristic. It has been conjectured that for any n, for p multilinear, the image of p evaluated on the set M n (K) of n by n matrices is either zero, or the set of scalar matrices, or the set sl n (K) of matrices of trace 0, or all of M n (K). This expository paper describes research on this problem and related areas. We discuss the solution of this conjecture for n = 2 in Section 2, some decisive results for n = 3 in Section 3, and partial information for n ≥ 3 in Section 4, also for non-multilinear polynomials. In addition we consider the case of K not algebraically closed, and polynomials evaluated on other finite dimensional simple algebras (in particular the algebra of the quaternions). This review recollects results and technical material of our previous papers, as well as new results of other researches, and applies them in a new context. This article also explains the role of the Deligne trick, which is related to some nonassociative cases in new situations, underlying our earlier, more straightforward approach. We pose some problems for future generalizations and point out possible generalizations in the present state of art, and in the other hand providing counterexamples showing the boundaries of generalizations.
Let p be a multilinear polynomial in several non-commuting variables with coefficients in an arbitrary field K. Kaplansky conjectured that for any n, the image of p evaluated on the set Mn(K) of n by n matrices is either zero, or the set of scalar matrices, or the set sln(K) of matrices of trace 0, or all of Mn(K). This conjecture was proved for n = 2 when K is closed under quadratic extensions. In this paper the conjecture is verified for K = R and n = 2, also for semi-homogeneous polynomials p, with a partial solution for an arbitrary field K.
Abstract. Kaplansky asked about the possible images of a polynomial f in several noncommuting variables. In this paper we consider the case of f a Lie polynomial. We describe all the possible images of f in M 2 (K) and provide an example of f whose image is the set of non-nilpotent trace zero matrices, together with 0. We provide an arithmetic criterion for this case. We also show that the standard polynomial s k is not a Lie polynomial, for k > 2.
Let p be a multilinear polynomial in several non-commuting variables with coefficients in an arbitrary field K. Kaplansky conjectured that for any n, the image of p evaluated on the set Mn(K) of n by n matrices is a vector space. In this paper we settle the analogous conjecture for a quaternion algebra.
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