We obtain two equivalent conditions for m polynomials in n variables to form a p-basis of a ring of constants of some polynomial K-derivation, where K is a unique factorization domain of characteristic p > 0. One of these conditions involves Jacobians while the other some properties of factors. In the case m = n this extends the known theorem of Nousiainen, and we obtain a new formulation of the Jacobian conjecture in positive characteristic.
@article{bwmeta1.element.doi-10_2478_s11533-013-0207-y,
author = {Piotr J\k edrzejewicz},
title = {A characterization of p-bases of rings of constants},
journal = {Open Mathematics},
volume = {11},
year = {2013},
pages = {900-909},
zbl = {1309.13032},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-013-0207-y}
}
Piotr Jędrzejewicz. A characterization of p-bases of rings of constants. Open Mathematics, Tome 11 (2013) pp. 900-909. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-013-0207-y/
[1] Adjamagbo K., On separable algebras over a U.F.D. and the Jacobian conjecture in any characteristic, In: Automorphisms of Affine Spaces, Curaçao, July 4-8, 1994, Kluwer, Dordrecht, 1995, 89–103 | Zbl 0832.13017
[2] van den Essen A., Polynomial Automorphisms and the Jacobian Conjecture, Progr. Math., 190, Birkhäuser, Basel, 2000 | Zbl 0962.14037
[3] van den Essen A., Nowicki A., Tyc A., Generalizations of a lemma of Freudenburg, J. Pure Appl. Algebra, 2003, 177(1), 43–47 http://dx.doi.org/10.1016/S0022-4049(02)00175-5 | Zbl 1040.13005
[4] Freudenburg G., A note on the kernel of a locally nilpotent derivation, Proc. Amer. Math. Soc., 1996, 124(1), 27–29 http://dx.doi.org/10.1090/S0002-9939-96-03003-1 | Zbl 0857.13005
[5] 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
[6] Jędrzejewicz P., Eigenvector p-bases of rings of constants of derivations, Comm. Algebra, 2008, 36(4), 1500–1508 http://dx.doi.org/10.1080/00927870701869014 | Zbl 1200.13040
[7] Jędrzejewicz P., One-element p-bases of rings of constants of derivations, Osaka J. Math., 2009, 46(1), 223–234 | Zbl 1159.13014
[8] 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 | Zbl 1216.13017
[9] Jędrzejewicz P., A note on rings of constants of derivations in integral domains, Colloq. Math., 2011, 122(2), 241–245 http://dx.doi.org/10.4064/cm122-2-9
[10] 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
[11] Jędrzejewicz P., A characterization of Keller maps, J. Pure Appl. Algebra, 2013, 217(1), 165–171 http://dx.doi.org/10.1016/j.jpaa.2012.06.015 | Zbl 1267.14077
[12] Matsumura H., Commutative Algebra, 2nd ed., Math. Lecture Note Ser., 56, Benjamin/Cummings, Reading, 1980
[13] Niitsuma H., Jacobian matrix and p-basis, TRU Math., 1988, 24(1), 19–34 | Zbl 0716.13005
[14] Niitsuma H., Jacobian matrix and p-basis, In: Topics in Algebra, Banach Center Publ., 26(2), PWN, Warszawa, 1990, 185–188
[15] Nousiainen P.S., On the Jacobian Problem, PhD thesis, Pennsylvania State University, 1982
[16] Nowicki A., Polynomial Derivations and their Rings of Constants, Habilitation thesis, Nicolaus Copernicus University, Torun, 1994, available at http://www-users.mat.umk.pl/_anow/ps-dvi/pol-der.pdf | Zbl 1236.13023
[17] 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
[18] 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
[19] Ono T., A note on p-bases of a regular affine domain extension, Proc. Amer. Math. Soc., 2008, 136(9), 3079–3087 http://dx.doi.org/10.1090/S0002-9939-08-09338-6 | Zbl 1153.13009