Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On the density of the set of generators of a polynomial algebra

Authors: Vesselin Drensky, Vladimir Shpilrain and Jie-Tai Yu
Journal: Proc. Amer. Math. Soc. 128 (2000), 3465-3469
MSC (1991): Primary 13B25; Secondary 16W20
Published electronically: June 7, 2000
MathSciNet review: 1690985
Full-text PDF

Abstract | References | Similar Articles | Additional Information


Let $K[X] = K[x_1,...,x_n], ~n \ge 2,$ be the polynomial algebra over a field $K$ of characteristic $0$. We call a polynomial $~p \in K[X]$ coordinate (or a generator) if $K[X] = K[p, p_2, ..., p_n]$ for some polynomials $~p_2, ..., p_n$. In this note, we give a simple proof of the following interesting fact: for any polynomial $~h~$ of the form $~(x_i + q),$ where $q$ is a polynomial without constant and linear terms, and for any integer $~m \ge 2$, there is a coordinate polynomial $~p~$ such that the polynomial $(p-h)$ has no monomials of degree $\leq m$. A similar result is valid for coordinate $k$-tuples of polynomials, for any $k < n$. This contrasts sharply with the situation in other algebraic systems.

On the other hand, we establish (in the two-variable case) a result related to a different kind of density. Namely, we show that given a non-coordinate two-variable polynomial, any sufficiently small perturbation of its non-zero coefficients gives another non-coordinate polynomial.

References [Enhancements On Off] (What's this?)

  • 1. D. J. Anick, Limits of tame automorphisms in $k[x_1,\ldots,x_N]$, J. Algebra 82 (1983), 459-468. MR 85d:13005
  • 2. P. M. Cohn, Subalgebras of free associative algebras, Proc. London Math. Soc. (3) 14 (1968), 618-632. MR 29:4777
  • 3. P. M. Cohn, Free rings and their relations, Academic Press, 1985. MR 87e:16006
  • 4. V. Drensky, Tame primitivity for free nilpotent algebras, C.R. Math. Rep. Acad. Sci. Canada (Math. Reprorts of the Acad. of Sci.) 14 (1992), 19-24. MR 93b:16045
  • 5. V. Drensky, Endomorphisms and automorphisms of relatively free algebras, Suppl. ai Rend. Circ. Mat. Palermo 31 (1993), 97-132. MR 94j:16063
  • 6. V. Shpilrain and J.-T. Yu, On generators of polynomial algebras in two commuting or non-commuting variables, J. Pure Appl. Algebra 132 (1998), 309-315. CMP 99:01

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 13B25, 16W20

Retrieve articles in all journals with MSC (1991): 13B25, 16W20

Additional Information

Vesselin Drensky
Affiliation: Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Akad. G. Bonchev Str., Block 8, 1113 Sofia, Bulgaria

Vladimir Shpilrain
Affiliation: Department of Mathematics, University of Hong Kong, Pokfulam Road, Hong Kong
Address at time of publication: Department of Mathematics, The City College, City University of New York, New York, New York 10027

Jie-Tai Yu
Affiliation: Department of Mathematics, University of Hong Kong, Pokfulam Road, Hong Kong

Received by editor(s): March 2, 1998
Received by editor(s) in revised form: February 22, 1999
Published electronically: June 7, 2000
Additional Notes: The first author was partially supported by Grant MM605/96 of the Bulgarian Foundation for Scientific Research.
The third author was partially supported by RGC-Fundable Grant 344/024/0004.
Communicated by: Wolmer V. Vasconcelos
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society