Irreducible Jacobian derivations in positive characteristic

Piotr Jędrzejewicz

Open Mathematics (2014)

  • Volume: 12, Issue: 8, page 1278-1284
  • ISSN: 2391-5455

Abstract

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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.

How to cite

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Piotr Jędrzejewicz. "Irreducible Jacobian derivations in positive characteristic." Open Mathematics 12.8 (2014): 1278-1284. <http://eudml.org/doc/269064>.

@article{PiotrJędrzejewicz2014,
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.},
author = {Piotr Jędrzejewicz},
journal = {Open Mathematics},
keywords = {Jacobian derivation; Ring of constants; p-basis; ring of constants; -basis},
language = {eng},
number = {8},
pages = {1278-1284},
title = {Irreducible Jacobian derivations in positive characteristic},
url = {http://eudml.org/doc/269064},
volume = {12},
year = {2014},
}

TY - JOUR
AU - Piotr Jędrzejewicz
TI - Irreducible Jacobian derivations in positive characteristic
JO - Open Mathematics
PY - 2014
VL - 12
IS - 8
SP - 1278
EP - 1284
AB - 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.
LA - eng
KW - Jacobian derivation; Ring of constants; p-basis; ring of constants; -basis
UR - http://eudml.org/doc/269064
ER -

References

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  2. [2] Freudenburg G., Algebraic Theory of Locally Nilpotent Derivations, Encyclopaedia Math. Sci., 136, Springer, Berlin, 2006 Zbl1121.13002
  3. [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 Zbl1024.13008
  4. [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 Zbl1118.13027
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  7. [7] Jędrzejewicz P., Jacobian conditions for p-bases, Comm. Algebra, 2012, 40(8), 2841–2852 http://dx.doi.org/10.1080/00927872.2011.587213 Zbl1254.13028
  8. [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. [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. [10] Matsumura H., Commutative Algebra, 2nd ed., Math. Lecture Note Ser., 56, Benjamin/Cummings, Reading, 1980 
  11. [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 Zbl1236.13023
  12. [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 Zbl0665.12024
  13. [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 Zbl0934.13001

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