Do we need to state theorems in an exam, or can we just use them?
As a general rule, if questions say something to the effect "stating carefully any results used" then you would get marks for stating it - or lose marks for not doing so. I don't think my course is any different from others in that respect.
If it doesn't say that, then it is enough to show you understand the theorem by saying, for example, \(\phi\) is a diffeomorphism near 0 by the inverse function theorem, because its Jacobian at 0 is invertible. (Rather than just saying, "by the inverse function theorem".)
What does \(\mathcal{E}_n^2\) mean?
- It's defined to be
How are the \(\mathcal{R}\)- and \(\mathcal{K}\)- finite determinacy theorems related
In both cases, the condition is \[\mathfrak{m}_n^{k+1}\theta(f) \ \subset \ \mathfrak{m}_n\,T\mathcal{G}\cdot f.\]
For \(\mathcal{R}\)-equivalence, \(T\mathcal{R}\cdot f = \mathfrak{m}_n\,Jf\), and recall that with p=1, \(\theta(f) \equiv \mathcal{E}_n\), so the left-hand-side is just \(\mathfrak{m}_n^{k+1}\).
Tangent space \(T\mathcal{K}\cdot f\)
Note that what I called \(Jf\) in the lectures is the same as \(tf(\theta(n))\) and is also denoted \(T\mathcal{R}_e\cdot f\).
Similarly, \(\mathfrak{m}_n\,Jf = tf(\mathfrak{m}_n\theta(n)) = T\mathcal{R}\cdot f\).
The example I did in lectures \(f(x,y) = (x^2,\,y^2)\) is also done in Chapter 11 (see p.114). For another example, see here.
I'm just not too sure what 'parametrizing \(C_F\) by (x,y,u)' means. Do we only keep the (x,y,u) terms?
Remember (1st year?) spherical polar coordinates: varying \(\theta\) and \(\phi\) give different points on the sphere (r = contant). That is a parametrization of the sphere by \(\theta\) and \(\phi\) (one has \(x= \sin\theta\, \cos\phi\) etc). Similarly, here if we vary x,y,u, by putting v = (function of x,y,u) (much like \(x=\sin \theta\) etc) we get different points on \(C_F\). So that is a parametrization of \(C_F\).
If one needs to do any calculations on the sphere or on \(C_F\), it is enough to do them using the parameters: eg finding the points where \(\pi_F\) is not a local diffeo - just write \(\pi_F\) in terms of \(x,y,u\) (and find its Jacobian etc).
What is a cobasis?
Definition A cobasis of I in \(\mathcal{E}_n\) is a set of functions (usually chosen to be monomials) \(h_1,h_2,\dots,h_k\) such that given any function \(f\in\mathcal{E}_n\) there are unique real numbers (coefficients) \(a_1,\dots,a_k\) and a function \(g\in I\) such that \[f = a_1h_1+\cdots + a_kh_k \ + \ g.\] (The uniqueness of the coefficients is what makes it a cobasis - otherwise they are just generators for \(\mathcal{E}_n/I\), see properties of basis in linear algebra).
If we want the cobasis for I in \(\mathfrak{m}_n\) instead, then replace \(\mathcal{E}_n\) by \(\mathfrak{m}_n\) in the definition above (in practice the difference is whether we include 1 in the cobasis or not).
Example Consider \(I=\left<xy,\,x^3+y^3\right> \subset \mathcal{E}_2\) (see Ex 3.9(ii)). There is a choice of cobasis. For example \(\{1,x,y,x^2,y^2,x^3\}\). Another is \(\{1,x,y,x^2,y^2,y^3\}\). But you can't have both \(x^3\) and \(y^3\), as if you did the coefficients would not be unique. For example, if you take \(f=2x^3+3y^2\), and you allow both you can write \[f = 2x^3+3y^2 + (0)\quad\mbox{as well as}\quad f = 3y^2-2y^3 + (2x^3+2y^3)\] (The term in brackets ( ) is the \(g\in I\).) More examples can be found in Exercise 3.9.
Further discussion: Consider the quotient \(\mathcal{E}_n/I\) (which is a vector space). A cobasis for an ideal \(I\) in \(\mathcal{E}_n\) is essentially a basis for this quotient. However, strictly speaking, the elements of \(\mathcal{E}_n/I\) are not functions but are of the form \(f+I\) where \(f\in\mathcal{E}_n\), rather than just \(f\); one says \(f\) is a representative of \(f+I\). Then if \(h_1,h_2,\dots,h_k\) are representatives such that \[\{h_1+I,\,h_2+I,\,\dots,\,h_k+I\}\] is a basis for \(\mathcal{E}_n/I\), then \(\{h_1,h_2,\dots,h_k\}\) is a cobasis of \(I\) in \(\mathcal{E}_n\).
Calculations with ideals (07/11/09)
Several people asked me questions about this over the last couple of days.
Why is \(\left<2x^2,\,3y^2\right> = \left<x^2,\,y^2\right>\), say?
The answer is simply that since 1/2 is in the ring of germs \(\mathcal{E}_2\) (it's a constant function), if \(2x^2\in I\) then it follows that \((1/2)2x^2\in I\) (definition of ideal: \(h\in I, k\in R \Rightarrow kh\in I\)). Similarly, \(y^2 = (1/3)(3y^2)\).
On the other hand, one cannot simplify say \(\left<x+3y^2,\,xy^3\right>\) - it is not equal to \(\left<x+y^2,\,xy^3\right>\).
How do we choose a basis for \(\mathcal{E}_2/Jf\) when Jf is not generated by monomials?
First find as many monomials as possible in Jf. Then with the rest make sure you don't choose elements for the basis, where a linear combination of them is in the ideal.
For example, if \(Jf = \left<x^2+y^2,\,xy\right>\) then the monomials in the ideal are \(x^3,\, xy,\, y^3\) (I hope you agree). The monomials not in \(Jf\) are therefore \(1,\,x,\, y,\,x^2,\,y^2\).. However, the sum of the last two is in \(Jf\), so they are not both needed - but you must include one of them, so one basis for \(\mathfrak{m}/Jf\) is \(\{x,\,y,\,x^2\}\). Or instead of \(x^2\) we could use \(y^2\), or \(x^2-y^2\), or any other linear combination of the two except (multiples of) \(x^2+y^2\).