AIMS Mathematics, 2020, 5(4): 3547-3555. doi: 10.3934/math.2020230

Research article

Export file:

Format

  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text

Content

  • Citation Only
  • Citation and Abstract

On the Tame automorphisms of differential polynomial algebras

Department of Mathematics, Harran University, Şanlıurfa, Turkey

Let $R\{x,y\}$ be the differential polynomial algebra in two differential indeterminates $x,y$ over a differential domain $R$ with a derivation operator $\delta$. In this paper, we study on automorphisms of the differential polynomial algebra $R\{x,y\}$ with one derivation operator. Using a method in group theory, we prove that the Tame subgroup of automorphism of $R\{x,y\}$ is the amalgamated free product of the Triangular and the Affine subgroups over their intersection.
  Figure/Table
  Supplementary
  Article Metrics

References

1. M. Aschenbrenner, L. Van Den Dries, J. Van Der Hoeven, Asymptotic Differential Algebra and Model Theory of Transseries, Princeton University Press, 2017.

2. A. G. Czerniakiewicz, Automorphisms of a free associative algebra of rank 2. I & II, T. Am. Math. Soc., 160 (1971), 393-401; 171 (1972), 309-315.

3. P. M. Cohn, Subalgebras of free associative algebras, P. Lond. Math. Soc., 56 (1964), 618-632.

4. B. A. Duisengaliyeva, A. S. Naurazbekova, U. U. Umirbaev, Tame and wild automorphisms of differential polynomial algebras of rank 2, Fund. Appl. Math., 22 (2019), 101-114.

5. G. Gallo, B. Mishra, F. Ollivier, Some constructions in rings of differential polynomials, In: Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, Springer, Berlin, 1991, 171-182.

6. H. W. E. Jung, Uber ganze birationale Transformationen der Ebene, J. Reine Angew. Math., 184 (1942), 161-174.

7. I. Kaplansky, An Introduction to Differential Algebra, Hermann, Paris, 1957.

8. E. R. Kolchin, Differential Algebra and Algebraic Groups, Academic Press, New York, 1973.

9. W. van der Kulk, On polynomial rings in two variables, Nieuw Archief voor Wiskunde, 3 (1953), 33-41.

10. L. Makar-Limanov, The automorphisms of the free algebra with two generators, Funct. Anal. Appl., 4 (1970), 262-263.

11. M. Nagata, On the Automorphism Group of k[x,y], Kinokuniya, Tokio, 1972.

12. J. F. Ritt, Differential Algebra, American Mathematical Society, New York, 1950.

13. I. P. Shestakov, U. U. Umirbaev, Tame and wild automorphisms of rings of polynomials in three variables, J. Am. Math. Soc., 17 (2004), 197-227.    

14. W. Sit, The Ritt-Kolchin Theory for Differential Polynomials, World Scientific, Singapore, 2001.

15. U. U. Umirbaev, The Anick automorphism of free associative algebras, J. Reine Angew. Math., 605 (2007), 165-178.

16. A. Van den Essen, Polynomial Automorphisms: and the Jacobian Conjecture, Birkhauser, 2012.

© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Article outline

Show full outline
Copyright © AIMS Press All Rights Reserved