An old problem (recently ``reheated" by Casas-Alvero, etc.) consists of trying to determine if degree $n$ polynomials
$P(t)$ that are $n$-th powers
are characterized by the following condition:


$$
\deg( gcd(P(t),P^{(r)}(t))) >0
$$
for all positive integers $r$ and $r \neq n,$
where $P^{(r)}(t)$ is the $r$-th formal derivative of $P(t)$ relative to $t$.


Seems that the case when the field of coefficients is the prime field $GF(p)$ has not been considered.


By chance I got the following polynomial $P\_{37}(t) \in GF(37)[t]$ that proves the (conjectural) result false:


$$
P\_{37}(t) = 17t^{12}+21t^4+24t^3+12t^2 \in GF(37)[t]
$$


We have indeed:


$$
\deg(\gcd(P\_{37}(t),P\_{37}^{(r)}(t)) \geq 1
$$
in $GF(37)[t]$
for all $r=1,2, \ldots,11$ and trivially for $r>13.$
But,


$$
P\_{37}(t)
$$
is not a $12$-th power in $GF(37)[t].$


More precisely:
$$
\gcd(P(t),P^{'}(t))=t, \quad \gcd(P(t),P^{''}(t)= t+36,
$$
$$
\gcd(P(t),P^{(3)}(t))=t+36,
\quad \gcd(P,P^{(4)}(t)=t+36,
$$
and all other $\gcd(P(t),P^{(r)}(t))=t$ for $r=5, \ldots,11$ and for $r>12.$


Question: There are other such examples ?. Can be characterized ?



Characterizing degree $n$ polynomials that are $n$-th powers in $GF(p)[t]$
$x^{p-1}-x^{(p-1)/2}$ for $p \equiv 1 \mod 4$.


