certain reduction argument that allows us to reduce successively the degree of the polynomial $P$ satisfying the conditions of Theorem \[main2\].

Non-microstates free entropy and free entropy dimension {#subsec:definitions}
-------------------------------------------------------

We want to catch up now on the definition of the non-microstates free entropy dimension.

Let $(M,\tau)$ be a tracial $W^\ast$-probability space and let self-adjoint elements $X_1,\dots,X_n\in M$ be given.

By possibly enlarging $(M,\tau)$, we may always assume that $(M,\tau)$ contains additionally semicircular elements $S_1,\dots,S_n$ such that $$\{X_1,\dots,X_n\}, \{S_1\}, \dots, \{S_n\}$$ are freely independent. Indeed, this can be done by replacing $(M,\tau)$ by the free product $(M,\tau) \ast_{\mathbb{C}}(L({\mathbb{F}}_n),\tau_n)$ of $(M,\tau)$ with the free group factor $(L({\mathbb{F}}_n),\tau_n)$.

Following Voiculescu [@Voi-Entropy-V], we define the *non-microstates free entropy* $\chi^\ast(X_1,\dots,X_n)$ of $X_1,\dots,X_n$ by $$\begin{aligned}
\lefteqn{\chi^\ast(X_1,\dots,X_n)}\\
&\quad := \frac{1}{2} \int^\infty_0\Big(\frac{n}{1+t}-\Phi^\ast(X_1+\sqrt{t}S_1,\dots,X_n+\sqrt{t}S_n)\Big)\, dt + \frac{n}{2}\log(2\pi e).\end{aligned}$$ We need to note that the function $$t\mapsto \Phi^\ast(X_1+\sqrt{t}S_1,\dots,X_n+\sqrt{t}S_n)$$ is well-defined, since [@Voi-Entropy-V Corollary 3.9] tells us that there exists a conjugate system of $(X_1+\sqrt{t}S_n,\dots,X_n+\sqrt{t}S_n)$ for all $t>0$. Moreover, we have the inequalities (cf. [@Voi-Entropy-V Corollary 6.14]) $$\label{eq:Fisher_perturbation}
\frac{n^2}{C^2 + nt} \leq \Phi^\ast(X_1+\sqrt{t}S_1,\dots,X_n+\sqrt{t}S_n) \leq \frac{n}{t} \quad\text{for all $t>0$},$$ where $C^2 := \tau(X_1^2 + \dots + X_n^2)$. The left inequality in particularly implies that (cf. [@Voi-Entropy-V Proposition 7.2]) $$\chi^\ast(X_1,\dots,X_n) \leq \frac{n}{2}\log(2\pi e 