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

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Abstract | References | Similar Articles | Additional Information

Let be the polynomial algebra over a field of characteristic . We call a polynomial *coordinate* (or a generator) if for some polynomials . In this note, we give a simple proof of the following interesting fact: for any polynomial of the form where is a polynomial without constant and linear terms, and for any integer , there is a coordinate polynomial such that the polynomial has no monomials of degree . A similar result is valid for *coordinate -tuples* of polynomials, for any . 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.

**1.**David J. Anick,*Limits of tame automorphisms of 𝑘[𝑥₁,\cdots,𝑥_{𝑁}]*, J. Algebra**82**(1983), no. 2, 459–468. MR**704764**, 10.1016/0021-8693(83)90160-6**2.**P. M. Cohn,*Subalgebras of free associative algebras*, Proc. London Math. Soc. (3)**14**(1964), 618–632. MR**0167504****3.**P. M. Cohn,*Free rings and their relations*, 2nd ed., London Mathematical Society Monographs, vol. 19, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1985. MR**800091****4.**Vesselin Drensky,*Tame primitivity for free nilpotent algebras*, C. R. Math. Rep. Acad. Sci. Canada**14**(1992), no. 1, 19–24. MR**1157269****5.**Vesselin Drensky,*Endomorphisms and automorphisms of relatively free algebras*, Rend. Circ. Mat. Palermo (2) Suppl.**31**(1993), 97–132. Recent developments in the theory of algebras with polynomial identities (Palermo, 1992). MR**1244615****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**

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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