Let $k$ be a field.
If $f \in k[x]$ is a polynomial, and $d/dx\ f = 0$, then either


1. $f$ is constant, or
2. $char\ k = p$ and $f \in k[x^p]$.


So "annihilated by all derivations" is perhaps not the right thing to
ask for in characteristic $p$ (though that's what I asked for in
[Is the singular locus ideal preserved by all derivations?](https://mathoverflow.net/questions/71145/is-the-singular-locus-ideal-preserved-by-all-derivations) ).



> 
> What is the right thing to ask for?
> 
> 
> 


I would like an invariance condition one could state of a subscheme
$Y$ of $X$, that holds for the singular locus, but doesn't hold for (say) any regular closed point on a rational variety. 



Replacement for derivations in characteristic p?
In principle, there are two possible approaches. One is based on the Hasse derivatives (also called hyperdifferentiations). See <http://math.fontein.de/2009/08/12/the-hasse-derivative/> for elementary definitions and properties, and the paper


P. Vojta, Jets via Hasse-Schmidt derivations, ArXiv: math/0407113,


for the use of Hasse derivatives in algebraic geometry.


On the analysis level, there are also the Carlitz derivatives, special difference operators working efficiently just on functions annihilated by usual derivatives, with a rich theory of "differential equations", Fourier series, special functions etc. See


A. N. Kochubei, Analysis in Positive Characteristic, Cambridge University Press, 2009.


However there is no geometry around this approach so far. 


