Have you any you (or has anybody else) a handle on what Bertram calls "Jordan theory", apparently some kind of algebra for representing such transformations? It looks powerful stuff, but I'd run out of puff by page 3 ! See eg. http://molle.fernuni-hagen.de/~loos/jordan/archive/gpg1/index.html http://molle.fernuni-hagen.de/~loos/jordan/archive/gpg2/index.html WFL On 8/29/14, Fred Lunnon <fred.lunnon@gmail.com> wrote:
That's a very neat construction! WFL
On 8/28/14, Adam P. Goucher <apgoucher@gmx.com> wrote:
Yes, the degree is unbounded.
Let's say that an algebraic curve has 'crossing number' k if we can find a straight line which intersects it at K distinct points, where K is finite and K >= k.
In particular, by Bezout's theorem, crossing number <= degree.
Now, I will describe a composition of transformations capable of turning a curve C of crossing number k into a curve of crossing number 2k.
1. Find a line L1 which intersects C in (at least) k distinct points.
2. Find another such line L2 with that property, which is not parallel to L1 (e.g. by slightly perturbing L1).
3. Slightly perturb this pair of lines to produce a hyperbola H which intersects C in at least 2k distinct points.
4. Apply a projective transformation to turn H into a circle H'.
5. Apply a Möbius transformation to turn H' into a line H''.
The image of C under this pair of maps thus has crossing number 2k.
Consequently, we can 'boost' the degree of an algebraic curve arbitrarily high using projective transformations and Möbius maps.
Sincerely,
Adam P. Goucher
Sent: Thursday, August 28, 2014 at 11:23 PM From: "Fred Lunnon" <fred.lunnon@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] Stupid question about geometrical transformations
"represent these things" seems further to imply that the correspondence goes both ways, or at least that the degree of such functions is unbounded.
Is that the case? Where does one learn about such matters? WFL
On 8/28/14, Adam P. Goucher <apgoucher@gmx.com> wrote:
It's infinite-dimensional.
We can certainly represent these things as maps of the form:
(x, y, z) --> (f(x,y,z), g(x,y,z), h(x,y,z)) [for two dimensions; generalise as appropriate]
where f, g and h are homogeneous rational functions, where deg(numerator) = deg(denominator) + 1.
Sent: Thursday, August 28, 2014 at 8:34 PM From: "Fred Lunnon" <fred.lunnon@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] Stupid question about geometrical transformations
Oops --- should read (n+1)^2-1 + (n+2)(n+1)/2 - (n+1)n/2 - 1 = n^2 + 3n .
One immediate difficulty is incompatible compactifications --- a hyperplane versus a single point at infinity.
WFL
On 8/28/14, Fred Lunnon <fred.lunnon@gmail.com> wrote:
What is generated by the union of projective and conformal (Moebius) groups?
Since these two intersect in similarities, the super-group in n-space has dimension at least (n^2-1) + (n+2)(n+1)/2 - (n+1)n/2 - 1 = n^2 + n - 1 ; just how big is it?
How should such transformations be represented for computational purposes?
Why don't I know the answers to these apparently obvious questions? [Uh, maybe don't answer that one right now ...]
Physicists have previously devoted some thought to this matter: in particular, a promising paper by Wolfgang Bertram (2001) at http://www.emis.de/journals/AG/2-4/2_329.pdf launches into discussing "Jordan functors", which will however surely cost this innocent much gruesome effort to decode.
[Pascual Jordan certainly seems put himself about, despite which I don't recall ever having encountered him before this week.]
Fred Lunnon
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