Polynomial over a finite field is a composition of a separable polynomial and a $x^{p^e}$. In the proof of Corollary 3.2 (in the paper http://www.math.iitb.ac.in/~srg/preprints/CarlitzWan.pdf)
asserts the following:
Let $\mathbb{F}_q$ be the finite field of $q$ elements and $p=char(\mathbb{F}_q)$.
If $f \in \mathbb{F}_q[x]$, then $f(x)=g(x^{p^e})$ for some nonnegative integer $e$ and a separable polynomial $g \in \mathbb{F}_q[x]$.
I think this is not true: for example in the case $q=9$ and $f(x)=(x-2)^2(x-1)=x^3-2x^2+2x-4 \in \mathbb{F}_9[x]$. We have that $f(x)$ is not separable and cannot be written in the form $g(x^{3^e})$.
Even assuming the hypotheses $\mathrm{GCD}(n, q-1)>1$ and $q \ge n^2(n-2)^2$, this doesn't seem true.
Where am I going wrong?
Thanks in advance.
 A: The issue is that there are multiple different definitions of separable polynomials, and the authors of that paper did not indicate which one they meant.  Your example is correct for one of the usual definitions of separability taught in Algebra courses, but the authors had in mind a different definition.  In the literature on permutation polynomials and other topics involving mappings in positive characteristic, a nonconstant polynomial $f(x)\in K[x]$ is called separable if the field extension $K(x)/K(f(x))$ is separable.  Here are two equivalent formulations (these equivalences aren't too hard to prove):

*

*$f(x)$ is separable if and only if the polynomial $f(x)-t$ over
the field $K(t)$ has no multiple roots in any extension of $K(t)$ (i.e., $f(x)-t$ is separable in one of the usual
senses), where $t$ is transcendental over $K$.


*$f(x)$ is separable if and only if $f(x)$ is not in $K[x^p]$,
where $p$ is the characteristic of $K$.
This notion of a polynomial being separable is a special case of the notion in algebraic geometry of a map between curves (or varieties or schemes or stacks) being separable, with the map here being from the affine line to itself.  The idea is to factor out the "purely inseparable" part of the map, whose behavior is usually completely understood and not very interesting, in order to focus your attention on the separable part of the map, which is where most of the interesting things happen.
In the case of polynomials, certainly a nonconstant polynomial $f(x)\in\mathbb F_q[x]$ can be written in exactly one way as $f(x)=g(x^{p^e})$ with $e\ge 0$ and $g(x)\in\mathbb F_q[x]$ separable, where $p:=\text{char}(\mathbb F_q)$.  Since $x^p$ permutes $\mathbb F_q$, it follows that $f(x)$ permutes $\mathbb F_q$ if and only if $g(x)$ does.  The advantage of studying $g(x)$ rather than $f(x)$ is that separability of $g(x)$ means the field extension $\mathbb F_q(x)/\mathbb F_q(g(x))$ is separable, and hence one can use Galois theory to turn questions about this field extension into questions about finite groups.  For instance, Fried, Guralnick, and Saxl used this approach, combined with serious group theory, to prove the following amazing result:
if $g(x)$ is a separable permutation polynomial over $\mathbb F_q$ of degree $n$ where $q\ge n^4>1$, and $g(x)$ is indecomposable in the sense that it cannot be written as the composition of polynomials in $\mathbb F_q[x]$ of strictly lower degrees, then one of these holds, where $p:=\text{char}(\mathbb F_q)$:

*

*$n$ is a prime coprime to $p^2-p$

*$n$ is a power of $p$

*$n=p^k(p^k-1)/2$ where $p\in\{2,3\}$ and $k$ is odd with $k>1$.

Moreover, they determined a lot of further information about the Galois group of the Galois closure of $\mathbb F_q(x)/\mathbb F_q(g(x))$, which made it possible for researchers in the area to completely classify the examples in cases 1 and 3, and also to completely classify the examples in case 2 when $n$ is $p$, $p^2$, or $p^3$.  It's easy to show that every separable permutation polynomial of degree at least $2$ can be written as the composition of indecomposable separable permutation polynomials (sometimes in many different ways), so these results yield strong conclusions for arbitrary permutation polynomials.  What I have always found magical is that the known group theory results yield fantastic conclusions for any indecomposable separable polynomial, or more generally for the Galois group of the Galois closure of any separable degree-$n$ field extension which has no nontrivial intermediate fields (for instance, for most $n$'s this Galois group can only be $A_n$ or $S_n$ -- a result of Cameron, Neumann, and Teague).  Reducing to the separable case, and then to the indecomposable case (or the case of a "minimal" field extension), lets one apply the incredible power of modern group theory when studying various sorts of questions about polynomials or field extensions or topological maps or ...
Regarding the specific paper mentioned in the question, I note that stronger results than those in that paper have been known for decades, with simpler proofs.  I wrote a very brief survey of this topic at
http://dept.math.lsa.umich.edu/~zieve/papers/epfacts.pdf
and surveyed the analogous questions for rational functions (while also proving new results) in
https://arxiv.org/pdf/2010.15657.pdf
Please feel free to contact me with any questions about this topic, as it is one I'm quite fond of, but it has an enormous number of papers, so it's not easy for newcomers (and sometimes also not-so-newcomers) to identify the interesting and important questions in the area.
Footnote: One should always be careful to define what one means by separability of a polynomial, since some standard Algebra textbooks say a polynomial is separable if it has no multiple roots in any extension field, while others say a polynomial is separable if each of its irreducible factors has no multiple roots in any extension field.  For instance, $x^2$ is separable under the second definition but not the first.
