Re: Proving something is an affine variety
- From: "Rupert" <rupertmccallum@xxxxxxxxx>
- Date: 5 Jan 2007 17:43:31 -0800
ngr wrote:
Question: Prove that a single point (a1,...,an) E k^n is an affine variety.
How would I go about doing this? I can't really think of how to do it properly.
(a1,...,an) is an arbitrarily chosen point, and I believe that there must be a function that equals 0 at that point - a function that is also a zero polynomial would work, or a function that is just f(a1,...,an) = 0, the zero function. If there is at least one function f where f(a1,...,an)=0, then (a1,...,an) can be considered an affine variety, correct? I guess my biggest problem is I don't know how to prove that there will always be a function that is zero at (a1,...,an), other than stating f(a1,...,an)=0 is always a possible function.
Ideas? Suggestions to put me on the right track? Thanks!
You need to come up with a finite set of polynomials such that the only
point satisfying those polynomials is (a_1,...,a_n). This is easy:
x_1=a_1, x_2=a_2, ... x_n=a_n. Some definitions of affine variety also
require that affine varieties be irreducible, but irreducibility is
trivial in this case.
.
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- Proving something is an affine variety
- From: ngr
- Proving something is an affine variety
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