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 \(\mathfrak{m}_n^{k+1}\).
Problem sheet 2
You will have noticed that some of the solutions are missing. Those problems are for general understanding, and not specifically necessary for the syllabus.
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 posible 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) \(x^2+y^2\).