On the density of the set of generators of a polynomial algebra
Authors:
Vesselin Drensky, Vladimir Shpilrain and JieTai Yu
Journal:
Proc. Amer. Math. Soc. 128 (2000), 34653469
MSC (1991):
Primary 13B25; Secondary 16W20
Published electronically:
June 7, 2000
MathSciNet review:
1690985
Fulltext PDF Free Access
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Abstract: 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 twovariable case) a result related to a different kind of density. Namely, we show that given a noncoordinate twovariable polynomial, any sufficiently small perturbation of its nonzero coefficients gives another noncoordinate 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
JieTai Yu
Affiliation:
Department of Mathematics, University of Hong Kong, Pokfulam Road, Hong Kong
Email:
yujt@hkusua.hku.hk
DOI:
http://dx.doi.org/10.1090/S0002993900054484
PII:
S 00029939(00)054484
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 RGCFundable Grant 344/024/0004.
Communicated by:
Wolmer V. Vasconcelos
Article copyright:
© Copyright 2000
American Mathematical Society
