Question

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 http://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.

Answer 1

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.