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.
@article{bwmeta1.element.doi-10_2478_s11533-014-0402-5,
author = {Piotr J\k edrzejewicz},
title = {Irreducible Jacobian derivations in positive characteristic},
journal = {Open Mathematics},
volume = {12},
year = {2014},
pages = {1278-1284},
zbl = {1312.13032},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-014-0402-5}
}
Piotr Jędrzejewicz. Irreducible Jacobian derivations in positive characteristic. Open Mathematics, Tome 12 (2014) pp. 1278-1284. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-014-0402-5/
[1] Daigle D., On some properties of locally nilpotent derivations, J. Pure Appl. Algebra, 1997, 114(3), 221–230 http://dx.doi.org/10.1016/0022-4049(95)00173-5 | Zbl 0885.13003
[2] Freudenburg G., Algebraic Theory of Locally Nilpotent Derivations, Encyclopaedia Math. Sci., 136, Springer, Berlin, 2006 | Zbl 1121.13002
[3] Jędrzejewicz P., Rings of constants of p-homogeneous polynomial derivations, Comm. Algebra, 2003, 31(11), 5501–5511 http://dx.doi.org/10.1081/AGB-120023970 | Zbl 1024.13008
[4] Jędrzejewicz P., On rings of constants of derivations in two variables in positive characteristic, Colloq. Math., 2006, 106(1), 109–117 http://dx.doi.org/10.4064/cm106-1-9 | Zbl 1118.13027
[5] Jędrzejewicz P., Eigenvector p-bases of rings of constants of derivations, Comm. Algebra, 2008, 36(4), 1500–1508 | Zbl 1200.13040
[6] Jędrzejewicz P., A characterization of one-element p-bases of rings of constants, Bull. Pol. Acad. Sci. Math., 2011, 59(1), 19–26 http://dx.doi.org/10.4064/ba59-1-3
[7] Jędrzejewicz P., Jacobian conditions for p-bases, Comm. Algebra, 2012, 40(8), 2841–2852 http://dx.doi.org/10.1080/00927872.2011.587213 | Zbl 1254.13028
[8] Jędrzejewicz P., A characterization of p-bases of rings of constants, Cent. Eur. J. Math., 2013, 11(5), 900–909 http://dx.doi.org/10.2478/s11533-013-0207-y
[9] Makar-Limanov L., Locally Nilpotent Derivations, a New Ring Invariant and Applications, lecture notes, Bar-Ilan University, 1998, available at http://www.math.wayne.edu/~lml/lmlnotes.dvi
[10] Matsumura H., Commutative Algebra, 2nd ed., Math. Lecture Note Ser., 56, Benjamin/Cummings, Reading, 1980
[11] 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 | Zbl 1236.13023
[12] 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 | Zbl 0665.12024
[13] 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 | Zbl 0934.13001