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The images of non-commutative polynomials evaluated on \(2\times 2\) matrices. (English) Zbl 1241.16017
The authors study the following important problem, reputedly raised by Kaplansky: Let \(p\) be a polynomial in the free associative algebra \(K\langle x_1,\dots,x_m\rangle\) over an arbitrary field \(K\). What is the image \(\text{Im}(p)\) of \(p\) evaluated on the algebra \(M_n(K)\) of \(n\times n\) matrices? An important case of this problem was formulated by L’vov: If \(p\) is multilinear, is the set of values of \(p\) on \(M_n(K)\) a vector space? The answer into affirmative is equivalent to the following conjecture: If \(p\) is multilinear, then its image on \(M_n(K)\) is either \(\{0\}\), the set \(K\) of scalar matrices, the set \(sl_n(K)\) of matrices of trace zero, or the whole algebra \(M_n(K)\).
The results in the paper under review handle the case of \(2\times 2\) matrices over a quadratically closed field \(K\) of any characteristic. The latter means that the field \(K\) contains all zeros of non-constant polynomials \(f(x)\in K[x]\) of degree \(\leq 2\deg p\). The main result is that over a quadratically closed field \(K\) the image of the multilinear polynomial \(p\) on \(M_2(K)\) is either \(\{0\}\), \(K\), \(sl_2(K)\), or \(M_2(K)\).
This result is a consequence of a stronger result which is of independent interest: For an \(m\)-tuple of integers \((w_1,\dots,w_m)\), the polynomial \(p(x_1,\dots,x_m)\) is semi-homogeneous of weighted degree \(d\) if for each monomial \(h\) in \(p\), taking \(d_j\) to be the degree of \(x_j\) in \(h\), it holds \(d_1w_1+\cdots+d_mw_m=d\). If the semi-homogeneous polynomial \(p(x_1,\dots,x_m)\) is evaluated on the algebra \(M_2(K)\) over a quadratically closed field \(K\), then \(\text{Im}(p)\) is either \(\{0\}\), \(K\), the set of all non-nilpotent matrices having trace 0, \(sl_2(K)\), or a dense subset of \(M_2(K)\) with respect to the Zariski topology.

MSC:
16R30 Trace rings and invariant theory (associative rings and algebras)
16R50 Other kinds of identities (generalized polynomial, rational, involution)
16S50 Endomorphism rings; matrix rings
12E05 Polynomials in general fields (irreducibility, etc.)
12E10 Special polynomials in general fields
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