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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 ?

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(1) Casas-Alvarez should be replaced by Casas-Alvero. (2) ArXiv math 0605090, page 4, Section 3 has to do with Hasse derivatives, not true derivatives. – Luis H Gallardo Jan 14 '11 at 4:38
Corrected reference; added details of $\gcd$'s. – Luis H Gallardo Jan 14 '11 at 9:01
The degree 6 polynomial at the end of Section 3 in the reference you give is still a counterexample with true derivatives. – dke Jan 14 '11 at 10:53
@dke: Right.!!! – Luis H Gallardo Jan 14 '11 at 11:44
The simplest similar example I can come up with by hand is $x^2(x+1)(x+3)\bmod 7$. – dke Jan 14 '11 at 12:28
up vote 4 down vote accepted

$x^{p-1}-x^{(p-1)/2}$ for $p \equiv 1 \mod 4$.

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