Kernel of generalized Weitzenb\

L.P.Bedratyuk

Let $k[X]{\colon{}=}k[x_1,\ldots ,x_n]$ be a polynomial ring over a field $k$ of characteristic zero. For a generalized Weitzenböck derivation $D$ of the ring $k[X],$ a description of the constant rings $k[X]^{\,D}$ and $k(X)^{\,D}$ is given.

1. Nowicki A. Polynomial derivation and their Ring of Constants, UMK, Torun,1994.

2. van den Essen A. Polynomial automorphisms and the Jacobian conjecture, Progress in Mathematics, 190, Basel, 2000.

3. Freudenburg G. A survey of counterexamples to Hilbert's fourteenth problem// Serdica Math. J. --- 2001. --- P.171--192.

4. Bedratyuk L. On complete system of invariants for the binary form of degree 7// J. Symb. Comput. --- 2007. --- V.42. --- P.935-947.

5. Бедратюк Л.П. Елементи Казиміра диференціювань кільця многочленів// Математичні студії. --- 2007. --- Т.27, № 2. --- С.115--119.

6. Daigle D., Freudenburg G. A conterexample to Hilbert's fourtheenth problem in dimension five// J. of Algebra. --- 1999. --- V.221. --- V.528--535.

7. Weitzenböck R. Über die Invarianten von linearen Gruppen// Acta Math. --- 1932. --- Bd.58. --- S.231--293.

8. Roberts M. The covariants of a binary quantic of the n-th degree// Quarterly J. Math. --- 1861. --- V.4. --- P.168--178.

9. Hilbert D. Theory of algebraic invariants. --- Cambridge University Press, 1993.

10. Gordan P. Invariantentheorie. --- Teubner, Leipzig, 1885-87; reprinted by Chelsea Publ. Co., 1987.

11. von Gall F. Das vollständige Formensystem der binären Form achter Ordnung// Math. Annalen. --- 1880. --- Bd.17. --- S.31--51; S.139--152.

12. Cerezo A. Tables des invariants algébriques et rationnels d'une matrice nilpotente de petite dimension/ Prépublications Mathématiques, Université de Nice, 146, 1987.

13. Tan L. An algorithm for explicit generators of the invariants of the basis $G_a$-action// Comm. in Algebra. --- 1989. --- V.17. --- P.565--572.

14. van den Essen A. An algorithm to compute the invariant ring of a $G_a$-action on an affine variety// J. Symbolic Computation. --- 1993. --- V.16. --- P.551--555.

15. Glenn O. Treatise on theory of invariants. --- Boston, 1915.

16. Newstead P. Invariants of pencils of binary cubics// Math. Proc. Cambridge Phil. Soc. --- 1981. --- V.89. --- P.201--209.

	Kernel of generalized Weitzenböck's derivation of polynomial ring (in Ukrainian)
Author	L.P.Bedratyuk bedratyuk@ief.tup.km.ua, leonid.uk@gmail.com Khmelnytskyi National University
Abstract	Let $k[X]{\colon{}=}k[x_1,\ldots ,x_n]$ be a polynomial ring over a field $k$ of characteristic zero. For a generalized Weitzenböck derivation $D$ of the ring $k[X],$ a description of the constant rings $k[X]^{\,D}$ and $k(X)^{\,D}$ is given.
Keywords	polynomial ring, generalized Weitzenböck derivation, constant ring
DOI	doi:10.30970/ms.29.2.115-120
Reference	1. Nowicki A. Polynomial derivation and their Ring of Constants, UMK, Torun,1994. 2. van den Essen A. Polynomial automorphisms and the Jacobian conjecture, Progress in Mathematics, 190, Basel, 2000. 3. Freudenburg G. A survey of counterexamples to Hilbert's fourteenth problem// Serdica Math. J. --- 2001. --- P.171--192. 4. Bedratyuk L. On complete system of invariants for the binary form of degree 7// J. Symb. Comput. --- 2007. --- V.42. --- P.935-947. 5. Бедратюк Л.П. Елементи Казиміра диференціювань кільця многочленів// Математичні студії. --- 2007. --- Т.27, № 2. --- С.115--119. 6. Daigle D., Freudenburg G. A conterexample to Hilbert's fourtheenth problem in dimension five// J. of Algebra. --- 1999. --- V.221. --- V.528--535. 7. Weitzenböck R. Über die Invarianten von linearen Gruppen// Acta Math. --- 1932. --- Bd.58. --- S.231--293. 8. Roberts M. The covariants of a binary quantic of the n-th degree// Quarterly J. Math. --- 1861. --- V.4. --- P.168--178. 9. Hilbert D. Theory of algebraic invariants. --- Cambridge University Press, 1993. 10. Gordan P. Invariantentheorie. --- Teubner, Leipzig, 1885-87; reprinted by Chelsea Publ. Co., 1987. 11. von Gall F. Das vollständige Formensystem der binären Form achter Ordnung// Math. Annalen. --- 1880. --- Bd.17. --- S.31--51; S.139--152. 12. Cerezo A. Tables des invariants algébriques et rationnels d'une matrice nilpotente de petite dimension/ Prépublications Mathématiques, Université de Nice, 146, 1987. 13. Tan L. An algorithm for explicit generators of the invariants of the basis $G_a$-action// Comm. in Algebra. --- 1989. --- V.17. --- P.565--572. 14. van den Essen A. An algorithm to compute the invariant ring of a $G_a$-action on an affine variety// J. Symbolic Computation. --- 1993. --- V.16. --- P.551--555. 15. Glenn O. Treatise on theory of invariants. --- Boston, 1915. 16. Newstead P. Invariants of pencils of binary cubics// Math. Proc. Cambridge Phil. Soc. --- 1981. --- V.89. --- P.201--209.
Pages	115-120
Volume	29
Issue	2
Year	2008
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

Kernel of generalized Weitzenböck's derivation of polynomial ring (in Ukrainian)