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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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