such zero divisors.

In particular, this result allows to conclude that the distribution of any non-trivial self-adjoint non-commutative polynomial in the generators does not have atoms, if the generators have full non-microstates free entropy dimension. Note that in a random matrix language this allows to conclude that the asymptotic eigenvalue distribution of polynomials in random matrices has, under the above assumption, no atomic parts. Questions on the absence of atoms for polynomials in non-commuting random variables (or for polynomials in random matrices) have been an open problem for quite a while. Only recently there was some progress on this in such generality; in [@Shlyakthenko-Skoufranis], Shlyakhtenko and Skoufranis showed that polynomials in free variables exhibit (under the assumption of no atoms for each of the variables) no atoms; here we give a vast generalization of this, by showing that not freeness is the crucial issue but maximality of free entropy dimension.

Existence of conjugate variables and absence of algebraic relations {#algebraic_relations}
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Let ${\mathbb{C}}\langle Z_1,\dots,Z_n\rangle$ be the $\ast$-algebra of non-commutative polynomials in $n$ self-adjoint (formal) variables $Z_1,\dots,Z_n$. For $j=1,\dots,n$, we denote by $\partial_j$ the non-commutative derivative with respect to $Z_j$, i.e. $\partial_j$ is the unique derivation $$\partial_j:\ {\mathbb{C}}\langle Z_1,\dots,Z_n\rangle \to {\mathbb{C}}\langle 