On page 7 of André Galligo's paper Deformation of Roots of Polynomials via Fractional Derivatives, we find the following statement:
Lemma 1. Let $f(x)$ be a polynomial of degree $n$, then $P_f (x, q) := \frac{1}{n!} x^q \,\Gamma(n − q)\operatorname{Diff}^q (f)$ is a polynomial in $x$ and $q$, of total degree $n$. Notice that, for an initial value $a \in \mathbb{R}$, the polynomial can be written in the basis formed by the powers of $(x − a)$, then one considers the fractional derivative operator $\operatorname{Diff}^q_a$ and the corresponding polynomial $P_{f,a}(x, q)$. Here we set $a = 0$. We decided that, since we focus on the polynomial roots of a fixed univariate polynomial $f$, it is convenient to choose for $a$ the mean of the complex roots of $f$. Equivalently, thanks to the famous Tchirnaus transformation, after a real translation on the abscissa $x$ we can assume that the coefficient of $x^{n−1}$ vanishes, hence the mean of the roots of $f$ is $0$. This last property is also true for all derivatives with integer orders of $f$.
Clearly if $x^{n-1}$ vanishes then the mean of the roots will be $0$ since the sum of roots will be zero by Vieta's Formulas.
However, I have no idea what the supposedly "famous Tchirnaus transformation" is, and I cannot find a single decent reference to it outside of this paper. Does anyone have an idea concerning what this transformation is?
A more common spelling is "Tschirnhaus."
See for example Tschirnhaus transfromation on Wikipedia