We prove that an irreducible polynomial derivation in positive characteristic is a Jacobian derivation if and only if there exists an (n-1)-element p-basis of its ring of constants. In the case of two variables we characterize these derivations in terms of their divergence and some nontrivial constants.
 Nowicki A., Polynomial Derivations and their Rings of Constants, Habilitation thesis, Nicolaus Copernicus University, Toruń, 1994, available at http://www-users.mat.umk.pl/~anow/ps-dvi/pol-der.pdf
 Nowicki A., Nagata M., Rings of constants for k-derivations in k[x
1, …, x
n], J. Math. Kyoto Univ., 1988, 28(1), 111–118
 Ono T., A note on p-bases of rings, Proc. Amer. Math. Soc., 2000, 128(2), 353–360 http://dx.doi.org/10.1090/S0002-9939-99-05029-7