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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
DOI: https://doi.org/10.1090/S0002-9939-00-05448-4
Published electronically: June 7, 2000
MathSciNet review: 1690985
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Abstract:

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.


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Additional Information

Vesselin Drensky
Affiliation: Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Akad. G. Bonchev Str., Block 8, 1113 Sofia, Bulgaria
Email: drensky@banmatpc.math.acad.bg

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
Email: shpil@hkusua.hku.hk, shpil@groups.sci.ccny.cuny.edu

Jie-Tai Yu
Affiliation: Department of Mathematics, University of Hong Kong, Pokfulam Road, Hong Kong
Email: yujt@hkusua.hku.hk

DOI: https://doi.org/10.1090/S0002-9939-00-05448-4
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

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