Show that, if F is a field with infinitely many elements, then f(x)=g(x) for all x in F implies that f=g as polynomials. Proof: A polynomial over a field F has a unique factorisation. Therefore for f(x)=g(x) then they must have the same zeros. This means that their factorisations are the same or there is non-zero or 1 a in F such that f(x)=a g(x) which is absurd.
@Alizter the factorizations exist in the algebraic closure, which for beginners in field theory is a sophisticated construction. there is a more low-tech proof using gauss lemma and considering zers (hence linear factors, hence degree) of f-g.
@UserX while it's true in a sense "almost all" polynomials are separable, when we study field theory and polynomial rings we do not think of it as a special case but just more generic data intrinsic to a polynomial
suppose f-g has degree n. pick n+1 distinct scalars a_1, ..., a_n+1. they must all be zeros of f-g. so (x-a_1)(x-a_2)...(x-a_n+1) is a factor of f-g, using the lemma I mentioned inductively. but this implies f-g has degree >n, a contradiction.
this is why it's important the base field is infinite: you can have f(a)=g(a) for all a in F but f(x)=/=g(x) if the field F is finite
in particular, x^p-x and the zero polynomial 0 are equal evaluated at every element of the field of p elements, but they are not the same polynomial
fyi, i accepted Robert Israel's answer to my quintic question. i remember you saying you might post that limit as a separate question, so let me know if you do
on a completely different note: any calc. of variations people here? i was trying to research something which i was sure i'd seen before, but haven't found anything directly like what i was looking for
Suppose I consider Dido's problem: With the perimeter held fixed, which closed plane curve encloses the largest possible area? The answer, of course, is a circle.
what i thought was that there was some way to make this precise, in the following sense: Can one construct an evolution equation for the perturbed curve (i.e. taking that as the initial condition) which will smoothly evolve to the circle?
I think Mike refers to the brand new rolling limit. It's not a uniform N per month limit, but something that depends on how previous questions were received.
I changed the denominator because the other one was wrong; if the limit is indeed 0 it doesn't matter (and asymptotics will only differ by a value of $1/32$) but this is still the one that actually counts natural density
kk. i may edit the original question to just say "hey, MikeMiller put up a question about a quantitative question about solvable v. unsolvable" rather than asking it directly, since i've already accepted an answer
@Rafflesiaarnoldii: my 'real' motivation was: given an initial plane curve (not a circle), can i define an evolution equation which preserves both area and perimeter?
the calculus of variations connection there is a lot less obvious, since there's no quantity i'm evidently extremalizing
i was more interested in finding an evolution equation which has two invariant quantities rather than just one
@TedShifrin So how many different notions of connection do I have? I already know about affine connections from Petersen's class, I'm now learning about these hip Ehresmann connections... are these the main two? The first for vector bundles and the latter for principal $G$-bundles? Am I about to learn about twenty different kinds of connection?
@Semiclassical Sorry, never saw such a thing; nor can I imagine how it could work. Preserving one parameter is done by running the standard curve-shortening flow, while simultaneously rescaling. There's a seminar talk on such flows tomorrow; I'll try to remember to ask the speaker.
the link to an evolution equation was b/c i was hoping to find a linkage to a known special function of some kind
mostly because the two-bump example, if somehow converted into something which wasn't locally isometric, seemed like it would awfully reminiscient of this:
Right, but it's quite geometric ... Works for any projective bundle or tangent bundle of projective submanifold. That's where projective connections come from.
BTW - on your $G \to G/H$ question: the tangent bundle of $G/H$ is an $H$-bundle under the adjoint action, and one gets this bundle structure from the adjoint action of $H$ on $\mathfrak g/\mathfrak h$... satisfied now? :)
I was just thinking about the fact that $\mathfrak h$ is a subalgebra, hence the adjoint action of $\mathfrak h$ is well-defined on $\mathfrak g/\mathfrak h$
I was more huhing about the cheating comment... (And I think you put an emoticon there, but those don't show on my screen)
Hi I'm reading DUdley's book and I find a very interesting exercise but I'm stuck: This says: Let $0\le f_n\to f$ and $\int f_n\to c>0$. Show that $\int \lim f_n\in [0,c]$ and by an example show that any value in $[0,c]$ can be taken.
The first part is nothing more than the application of Fatou's lemma.
BUt for the example I have problems.
any ideas?
The only counterexamples I can think are the trivial as $f_n= 1_{[n,n-1)}$, as $f_n \to 0$ $\int f=0\le \int f_n =1$ but I can't figure out one way to get all the values in between $[0,1]$ for example.
So earlier I was in here asking about finding the index of a fixed point of a vector field in $\mathbb{R}^2$, where $x' = f(x, y)$ and $y' = g(x, y)$. Rigorously, that is given by $\displaystyle \int_C w$, where $w$ is the $1$-form $\displaystyle \frac{fdg - gdf}{f^2 + g^2}$ and $C$ is a closed curve around the fixed point.
I've tried some examples for practice and things work out, but now my question is why? Why does this work? We have informally defined the index is something like this: At each point along the closed curve, the vector field makes a particular angle w.r.t. the x-axis. Then the index is the net change in this angle over a complete circuit on $C$ divided by $2\pi$.
It might please you to know there are a bunch of different ways of thinking about winding number. There's differential geometric (yours), complex analytic, topological... and I think each gives different insights.
I see on Wikipedia that the original 1-form generates the original de-Rham cohomology group of $\mathbb{R} \setminus \{0\}$, and all other closed-but-not-exact forms give multiples of the winding number when integrated. That's really fascinating actually.
@DanielFischer do you know an elementary proof for $\operatorname{int}\operatorname{cl}C \subset C$ when $C$ is a convex in a normed space ? I know the stronger inclusion $\operatorname{int}\operatorname{cl}C \subset \operatorname{int} C$ also holds but both seem hard to prove without any background on simplexes or separating hyperplanes
My superannuation fund promises to pay me 30000 p.a. for the rest of my life. Alternatively if offers to give me a lump sum of 350 000 and, if i reinvest with the fund, to pay 8% p.a. interest. Should i accept the lump sum payment? I hope to live for a very long time.
