A tracial Nullstellensatz. (Q2922873)

The authors prove an analog of Hilbert's Nullstellensatz for polynomials in the free associative algebra when evaluated on matrix algebras. Let \(F\) be an algebraically closed field of characteristic 0, and let \(M(F)=\bigcup_{n\geq 0}M_n(F)\). Denote by \(F\langle X\rangle\) the free associative algebra freely generated over \(F\) by the set of \(g\) variables \(x_1,\ldots,x_g\). The main theorem of the paper is the following. Let \(f_1,\ldots,f_r,f\in F\langle X\rangle\) be polynomials. Then the vanishing of the traces of \(f_1(A),\ldots,f_r(A)\), for every choice of \(n\) and for every \(g\)-tuple of \(n\times n\) matrices, implies that \(f(A)\) is a traceless matrix if and only if \(f\) is cyclically equivalent to a linear combination of \(f_1,\ldots,f_r\), or else some linear combination of the \(f_i\) is cyclically equivalent to a nonzero scalar in \(F\langle X\rangle\). Recall that two polynomials \(f\) and \(g\) are cyclically equivalent if \(f-g\) is a sum of commutators in \(F\langle X\rangle\).