Positive characteristic analogs of closed polynomials

Piotr Jędrzejewicz

Open Mathematics (2011)

  • Volume: 9, Issue: 1, page 50-56
  • ISSN: 2391-5455

Abstract

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The notion of a closed polynomial over a field of zero characteristic was introduced by Nowicki and Nagata. In this paper we discuss possible ways to define an analog of this notion over fields of positive characteristic. We are mostly interested in conditions of maximality of the algebra generated by a polynomial in a respective family of rings. We also present a modification of the condition of integral closure and discuss a condition involving partial derivatives.

How to cite

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Piotr Jędrzejewicz. "Positive characteristic analogs of closed polynomials." Open Mathematics 9.1 (2011): 50-56. <http://eudml.org/doc/269507>.

@article{PiotrJędrzejewicz2011,
abstract = {The notion of a closed polynomial over a field of zero characteristic was introduced by Nowicki and Nagata. In this paper we discuss possible ways to define an analog of this notion over fields of positive characteristic. We are mostly interested in conditions of maximality of the algebra generated by a polynomial in a respective family of rings. We also present a modification of the condition of integral closure and discuss a condition involving partial derivatives.},
author = {Piotr Jędrzejewicz},
journal = {Open Mathematics},
keywords = {Closed polynomial; Derivation; Ring of constants; closed polynomial; derivations; rings of constants},
language = {eng},
number = {1},
pages = {50-56},
title = {Positive characteristic analogs of closed polynomials},
url = {http://eudml.org/doc/269507},
volume = {9},
year = {2011},
}

TY - JOUR
AU - Piotr Jędrzejewicz
TI - Positive characteristic analogs of closed polynomials
JO - Open Mathematics
PY - 2011
VL - 9
IS - 1
SP - 50
EP - 56
AB - The notion of a closed polynomial over a field of zero characteristic was introduced by Nowicki and Nagata. In this paper we discuss possible ways to define an analog of this notion over fields of positive characteristic. We are mostly interested in conditions of maximality of the algebra generated by a polynomial in a respective family of rings. We also present a modification of the condition of integral closure and discuss a condition involving partial derivatives.
LA - eng
KW - Closed polynomial; Derivation; Ring of constants; closed polynomial; derivations; rings of constants
UR - http://eudml.org/doc/269507
ER -

References

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  1. [1] Arzhantsev I.V., Petravchuk A.P., Closed polynomials and saturated subalgebras of polynomial algebras, Ukrainian Math. J., 2007, 59(12), 1783–1790 http://dx.doi.org/10.1007/s11253-008-0037-4 Zbl1164.13302
  2. [2] Ayad M., Sur les polynômes f(X, Y) tels que K[f] est intégralement fermé dans K[X, Y], Acta Arith., 2002, 105(1), 9–28 http://dx.doi.org/10.4064/aa105-1-2 
  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., 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
  5. [5] Jędrzejewicz P., One-element p-bases of rings of constants of derivations, Osaka J. Math., 2009, 46(1), 223–234 Zbl1159.13014
  6. [6] Nowicki A., On the jacobian equation J(f, g) = 0 for polynomials in k[x, y], Nagoya Math. J., 1988, 109, 151–157 Zbl0642.13016
  7. [7] Nowicki A., Polynomial Derivations and their Rings of Constants, UMK, Toruń, 1994 Zbl1236.13023
  8. [8] 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
  9. [9] Płoski A., On the irreducibility of polynomials in several complex variables, Bull. Polish Acad. Sci. Math., 1991, 39(3–4), 241–247 Zbl0766.12001
  10. [10] Schinzel A., Polynomials with Special Regard to Reducibility, Encyclopedia Math. Appl., 77, Cambridge University Press, Cambridge, 2000 

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