Abstract
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.
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Daigle D., On some properties of locally nilpotent derivations, J. Pure Appl. Algebra, 1997, 114(3), 221–230
Freudenburg G., Algebraic Theory of Locally Nilpotent Derivations, Encyclopaedia Math. Sci., 136, Springer, Berlin, 2006
Jędrzejewicz P., Rings of constants of p-homogeneous polynomial derivations, Comm. Algebra, 2003, 31(11), 5501–5511
Jędrzejewicz P., On rings of constants of derivations in two variables in positive characteristic, Colloq. Math., 2006, 106(1), 109–117
Jędrzejewicz P., Eigenvector p-bases of rings of constants of derivations, Comm. Algebra, 2008, 36(4), 1500–1508
Jędrzejewicz P., A characterization of one-element p-bases of rings of constants, Bull. Pol. Acad. Sci. Math., 2011, 59(1), 19–26
Jędrzejewicz P., Jacobian conditions for p-bases, Comm. Algebra, 2012, 40(8), 2841–2852
Jędrzejewicz P., A characterization of p-bases of rings of constants, Cent. Eur. J. Math., 2013, 11(5), 900–909
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
Matsumura H., Commutative Algebra, 2nd ed., Math. Lecture Note Ser., 56, Benjamin/Cummings, Reading, 1980
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
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Jędrzejewicz, P. Irreducible Jacobian derivations in positive characteristic. centr.eur.j.math. 12, 1278–1284 (2014). https://doi.org/10.2478/s11533-014-0402-5
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DOI: https://doi.org/10.2478/s11533-014-0402-5