One-element $p$-bases of rings of constants of derivations
Piotr Jędrzejewicz
Source: Osaka J. Math. Volume 46, Number 1
(2009), 223-234.
Abstract
In this paper we present sufficient conditions and necessary conditions for a single element to form a $p$-basis of a ring of constants of a derivation. We consider some special cases, when these conditions are equivalent, and we analyze some counter-examples, when the equivalence does not hold.
First Page:
Show
Hide
Full-text: Open access
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.ojm/1235574045
Mathematical Reviews number (MathSciNet): MR2531147
Zentralblatt MATH identifier: 1159.13014
References
M. Ayad: Sur les polynômes $f(X,Y)$ tels que $K[f]$ est intégralement fermé dans $K[X,Y]$, Acta Arith. 105 (2002), 9--28.
Mathematical Reviews (MathSciNet): MR1933390
Digital Object Identifier: doi:10.4064/aa105-1-2
D. Daigle and G. Freudenburg: Locally nilpotent derivations over a UFD and an application to rank two locally nilpotent derivations of $k[X_1,\ldots,X_n]$, J. Algebra 204 (1998), 353--371.
Mathematical Reviews (MathSciNet): MR1624439
Zentralblatt MATH: 0956.13007
Digital Object Identifier: doi:10.1006/jabr.1998.7465
A. van den Essen, A. Nowicki and A. Tyc: Generalizations of a lemma of Freudenburg, J. Pure Appl. Algebra 177 (2003), 43--47.
Mathematical Reviews (MathSciNet): MR1948836
Zentralblatt MATH: 1040.13005
Digital Object Identifier: doi:10.1016/S0022-4049(02)00175-5
G. Freudenburg: A note on the kernel of a locally nilpotent derivation, Proc. Amer. Math. Soc. 124 (1996), 27--29.
Mathematical Reviews (MathSciNet): MR1285990
Zentralblatt MATH: 0857.13005
Digital Object Identifier: doi:10.1090/S0002-9939-96-03003-1
JSTOR: links.jstor.org
P. Jędrzejewicz: Rings of constants of $p$-homogeneous polynomial derivations, Comm. Algebra 31 (2003), 5501--5511.
Mathematical Reviews (MathSciNet): MR2005240
Zentralblatt MATH: 1024.13008
Digital Object Identifier: doi:10.1081/AGB-120023970
P. J\kedrzejewicz: On rings of constants of derivations in two variables in positive characteristic, Colloq. Math. 106 (2006), 109--117.
Mathematical Reviews (MathSciNet): MR2234740
Zentralblatt MATH: 1118.13027
Digital Object Identifier: doi:10.4064/cm106-1-9
P. J\k edrzejewicz: Eigenvector $p$-bases of rings of constants of derivations, Comm. Algebra 36 (2008), 1500--1508.
Mathematical Reviews (MathSciNet): MR2406603
Zentralblatt MATH: 05293086
Digital Object Identifier: doi:10.1080/00927870701869014
H. Matsumura: Commutative Algebra, second edition, Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980.
Mathematical Reviews (MathSciNet): MR575344
A. Nowicki: On the Jacobian equation $J(f,g)=0$ for polynomials in $k[x,y]$, Nagoya Math. J. 109 (1988), 151--157.
Mathematical Reviews (MathSciNet): MR931957
Zentralblatt MATH: 0642.13016
Project Euclid: euclid.nmj/1118780897
A. Nowicki: Polynomial Derivations and Their Rings of Constants, Nicolaus Copernicus University, Toruń, 1994.
A. Nowicki: Rings and fields of constants for derivations in characteristic zero, J. Pure Appl. Algebra 96 (1994), 47--55.
Mathematical Reviews (MathSciNet): MR1297440
Zentralblatt MATH: 0811.12003
Digital Object Identifier: doi:10.1016/0022-4049(94)90086-8
A. Nowicki and M. Nagata: Rings of constants for $k$-derivations in $k[x_1,\ldots,x_n]$, J. Math. Kyoto Univ. 28 (1988), 111--118.
Mathematical Reviews (MathSciNet): MR929212
Zentralblatt MATH: 0665.12024
Project Euclid: euclid.kjm/1250520561
T. Ono: A note on $p$-bases of rings, Proc. Amer. Math. Soc. 128 (2000), 353--360.
Mathematical Reviews (MathSciNet): MR1623048
Zentralblatt MATH: 0934.13001
Digital Object Identifier: doi:10.1090/S0002-9939-99-05029-7
JSTOR: links.jstor.org
O. Zariski and P. Samuel: Commutative Algebra, volume I, D. Van Nostrand Company, Inc., Princeton, New Jersey, 1958.
Mathematical Reviews (MathSciNet): MR90581
2012 © Osaka University and Osaka City University, Departments of Mathematics
Osaka Journal of Mathematics
