Let $T:\mathbb Q(x)\to \mathbb Q(x)$ be the operator of inverse logarithmic derivative, i.e. $$Tf=\frac{f}{f'}.$$ Define $$p\_n(x)=T^n\left(x-\frac{x^2}{2}\right).$$ Let $f\_n(x) \in \mathbb Z[x]$ be the numerator of $p\_n(x)$, taken with positive leading term. The sequence $D\_n$ of discriminants of our polynomials starts with
$$\begin{aligned}
D\_0&=4,\\ D\_1&=4,\\ D\_2&=4,\\ D\_3&=-256=2^8,\\ D\_4&=2^{32}5^2,\\ D\_5&=2^{128}5^6331^2, \\D\_6&=2^{512}5^{20}331^646599695357^2.\\
\end{aligned}
$$


Is it true that we always have $D\_n=\pm a\_n^2$ for some integer $a\_n$? What else can be said about the arithmetic properties of $a\_n$? For example, is it true that for all $n \geq 2$ we have $\nu\_2(D\_n)=2^{2n-3}$?



Discriminant of numerator of inverse logarithmic derivative operator iteration
First, let's make the substitution $x\to 1-x$, so the function $x-\frac12x^2$
becomes $\frac12(1-x^2)$. The key here is that it is an even function, and the
discriminant of even and odd polynomials are essentially squares. More precisely,
$$\begin{aligned}
 \text{Disc}\bigl( A(x^2) \bigr) &= \pm\text{Resultant}\bigl(A(x^2),2xA'(x^2)\bigr)\\
 &= \pm 2^{\deg(A)} \cdot A(0)\cdot \text{Disc}\bigl( A(x) \bigr)^2
\end{aligned}
$$
and
$$\begin{aligned}
 \text{Disc}\bigl( xA(x^2) \bigr) &= \pm\text{Resultant}\bigl(xA(x^2),A(x^2)+2x^2A'(x^2)\bigr)\\
 &= \pm A(0)\text{Resultant}\bigl(A(x^2),2x^2A'(x^2)\bigr)\\
 &= \pm 2^{\deg(A)} \cdot A(0)^3\cdot \text{Disc}\bigl( A(x) \bigr)^2.\\
\end{aligned}
$$


Next, define three operators on the ring of rational functions:
$$ T(f)=\frac{f}{f'},\quad D(f)=\frac{f'}{f},\quad I(f)=\frac{1}{f}. $$
The question asks about iterates of $T$, but note that $T=I\circ D$ and
$D\circ I=-D$, so
$$ T^n(f) = (I\circ D)^n(f) = I\circ(D\circ I)^{n-1}\circ D(f) = (-1)^{n-1}I\circ D^n. $$
So up to sign, the numerator and denominator of $T^n(f)$ are the reverse of those for $D^n(f)$,
so it's enough to look at the logarithmic derivative $D$. Finally, we note that
if we start with an even function, then $Df$ is odd, and likewise if we start with an odd function,
then $Df$ is odd. Hence if $f$ is an even (or odd) rational function, then for all $n\ge1$ we
have $D^nf=A\_n/B\_n$ with one of $A\_n$ and $B\_n$ odd and the other even.
From above, the absolute values of the discriminants of $A\_n$ and $B\_n$ are squares,
up to the power of $2$ and the $A(0)$ term. But it's easy to check that for the particular
function $\frac12(1-x^2)$, the power of $2$ will be even and the $A(0)$ term will be $\pm1$.


