The implicit function theorem and free algebraic sets (Q2790587)

Denoting by \(\mathbb{P}^d\) the algebra of free polynomials in \(d\) variables, the authors regard an element \(p\in\mathbb{P}^d\) as a function which can be evaluated on subsets of \(\mathbb{M}^{[d]}= \bigcup_{n=1}^\infty\mathbb{M}_n^d\), where \(\mathbb{M}_n\) is the set of \(n\times n\) matrices with complex coefficients. Fixing an invertible \(X_0\in\mathbb{M}_n\) and letting \(\mathcal{Y}=\{Y\in\mathbb{M}_n: aX_0^2+bX_0Y+cYX_0=0\}\), with \(b\neq-c\), the authors describe the elements of \(\mathcal{Y}\) commuting with \(X_0\). This result is extended to a general theorem about free algebraic sets defined by \(d-1\) polynomials in \(d\) variables. The main tool is an implicit function theorem for non-commutative functions, in domains of \(\mathbb{M}^{[d]}\).