A characterization of p-bases of rings of constants

Piotr Jędrzejewicz

Open Mathematics (2013)

  • Volume: 11, Issue: 5, page 900-909
  • ISSN: 2391-5455

Abstract

top
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.

How to cite

top

Piotr Jędrzejewicz. "A characterization of p-bases of rings of constants." Open Mathematics 11.5 (2013): 900-909. <http://eudml.org/doc/269270>.

@article{PiotrJędrzejewicz2013,
abstract = {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.},
author = {Piotr Jędrzejewicz},
journal = {Open Mathematics},
keywords = {Derivation; Ring of constants; p-basis; Jacobian conjecture; derivation; ring of constants; -basis},
language = {eng},
number = {5},
pages = {900-909},
title = {A characterization of p-bases of rings of constants},
url = {http://eudml.org/doc/269270},
volume = {11},
year = {2013},
}

TY - JOUR
AU - Piotr Jędrzejewicz
TI - A characterization of p-bases of rings of constants
JO - Open Mathematics
PY - 2013
VL - 11
IS - 5
SP - 900
EP - 909
AB - 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.
LA - eng
KW - Derivation; Ring of constants; p-basis; Jacobian conjecture; derivation; ring of constants; -basis
UR - http://eudml.org/doc/269270
ER -

References

top
  1. [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 Zbl0832.13017
  2. [2] van den Essen A., Polynomial Automorphisms and the Jacobian Conjecture, Progr. Math., 190, Birkhäuser, Basel, 2000 Zbl0962.14037
  3. [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 Zbl1040.13005
  4. [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 Zbl0857.13005
  5. [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 Zbl1024.13008
  6. [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 Zbl1200.13040
  7. [7] Jędrzejewicz P., One-element p-bases of rings of constants of derivations, Osaka J. Math., 2009, 46(1), 223–234 Zbl1159.13014
  8. [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 Zbl1216.13017
  9. [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. [10] 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
  11. [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 Zbl1267.14077
  12. [12] Matsumura H., Commutative Algebra, 2nd ed., Math. Lecture Note Ser., 56, Benjamin/Cummings, Reading, 1980 
  13. [13] Niitsuma H., Jacobian matrix and p-basis, TRU Math., 1988, 24(1), 19–34 Zbl0716.13005
  14. [14] Niitsuma H., Jacobian matrix and p-basis, In: Topics in Algebra, Banach Center Publ., 26(2), PWN, Warszawa, 1990, 185–188 
  15. [15] Nousiainen P.S., On the Jacobian Problem, PhD thesis, Pennsylvania State University, 1982 
  16. [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 Zbl1236.13023
  17. [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 Zbl0665.12024
  18. [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 Zbl0934.13001
  19. [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 Zbl1153.13009

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.