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On the density of the set of generators of a polynomial algebra |
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| Authors: | Vesselin Drensky Vladimir Shpilrain Jie-Tai Yu |
| Institution: | Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Akad. G. Bonchev Str., Block 8, 1113 Sofia, Bulgaria ; Department of Mathematics, University of Hong Kong, Pokfulam Road, Hong Kong ; Department of Mathematics, University of Hong Kong, Pokfulam Road, Hong Kong |
| Abstract: | Let 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. |
| Keywords: | |
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![$KX] = Kx_1,...,x_n], ~n \ge 2,$](http://www.ams.org/proc/2000-128-12/S0002-9939-00-05448-4/gif-abstract0/img1.gif){kind=small}
be the polynomial algebra over a field 
of characteristic 
. We call a polynomial ![$~p \in KX]$](http://www.ams.org/proc/2000-128-12/S0002-9939-00-05448-4/gif-abstract0/img4.gif){kind=small}
coordinate (or a generator) if ![$KX] = Kp, p_2, ..., p_n]$](http://www.ams.org/proc/2000-128-12/S0002-9939-00-05448-4/gif-abstract0/img5.gif){kind=small}
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.