I am a student studying algebra and cryptography.
I wonder below question is possible.
Can I make some polynomials $f(x)$ over finite field that all derivations $f^{(k)}(x)$ are distinct when x is fixed but picked randomly?
i.e. $\forall x \in F, \forall k \in \{ 0,...,n \}, \ where \ n \ is \ degree \ and \ F \ is \ finite \ field, \\ \exists f \ s.t. \ \forall f^{(k)}(x) \ are \ distinct \ when \ random \ x \ is \ fixed \ ?$
I think it will be impossible, but also wonder that is possible when the finite field or polynomial has some specific conditions...
If there exists some related works, please tell me..!!
The choice of $x$ doesn't matter, because if $g(X) = f(X-x)$, $g$ is a polynomial of the same degree as $f$ and $g^{(k)}(x) = f^{(k)}(0)$. So let's choose $x = 0$ for convenience. Take any prime $p > n$. Then you can choose the coefficients $a_0, \ldots, a_n$ of $f(X) = a_0 + \ldots + a_n X^n$ so that $f^{(k)}(0) = k!\; a_k$ are congruent mod $p$ to any chosen $n+1$ values in $\{0,1,\ldots,p-1\}$.