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mathwonk

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isnt that what taylor polynomials are for? [n.j. hicks, notes on differential geometry, lemma, page 6.]

oops, that is just the local solution. i presume you can globalize it without too much trouble, using partition of unity.

on second i am not so clear on this. at least the ideal is generated by dimM elements in the local ring of the point.

but it is not immediately clear to me that this is even true except at the level of germs.

but compactness is a very strong property.

oops, that is just the local solution. i presume you can globalize it without too much trouble, using partition of unity.

on second i am not so clear on this. at least the ideal is generated by dimM elements in the local ring of the point.

but it is not immediately clear to me that this is even true except at the level of germs.

but compactness is a very strong property.

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Hurkyl

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Hurkyl

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Well, try considering the case where we are only interested in polynomial functions on some coordinate chart -- IMHO that case should be fairly easy and quite suggestive. (If you have trouble seeing it, look at the one-dimensional case of polynomials overMy guess was that you can approximate a function by some polynomials,

That you can do this locally is almost built into the definition of a (finite-dimensional) vector bundle -- there is be an obvious spanning set (in fact, a basis!) for the module of vector fields on n-dimensional real space. Remember that the coefficients can beso if you could finitely generate vector fields

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Hurkyl

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The different problems have different

[tex]\mathfrak{m} \neq \mathfrak{m}_\infty \cdot C(M)[/tex]

(the right hand side is, of course, a subset of

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mathwonk

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i knew hurkyl was the man who would take this **** seriously.

just kidding!!!

just kidding!!!

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I cover [tex]M[/tex] with finite number of charts [tex]U_{k}[/tex] with maps [tex]\phi_{k}[/tex]. In [tex]\mathbb{R}^{n}[/tex], functions [tex]f_{i}=x_{i}-x_{0i}[/tex] generate my ideal (from Taylor series expansion, right?), so in [tex]U_{n}[/tex] functions [tex]\phi_{k}f_{i}[/tex] generate my ideal locally. So I have a finite family of generators [tex]f_{i,k}[/tex] and I want to make it global, so I take a partition of unity [tex]g_{k}[/tex] and put [tex]f_{n}=\sum{g_{i}f_{i,n}[/tex]. Is this correct?

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