/sci/ - Science & Math

okay, here are some ideas...

these are essentially two equations of the form

<span class="math">\sum_i c_i · d_i · z^{c_i}=0[/spoiler],

where <span class="math">d_i[/spoiler] is short for the numbers which only depend on the other equation.

That expression can be written as

<span class="math">z · \frac{d}{dz} \sum_i d_i ·z^{c_i}[/spoiler].

Okay, since the polygon covers the origin (0,0), at least one of the <span class="math">c_i[/spoiler] is negative and one is positive. Therefore, the polynomial <span class="math">\sum_i d_i ·z^{c_i}[/spoiler] diverges for 0 and infinity. So, with positive d's (<span class="math">d_i=y^{b_i}[/spoiler]), the expression certainly has at least one minimum <span class="math">Z[/spoiler].

Then the derivative above is zero at Z or put differently <span class="math">\sum_i c_i · d_i · Z^{c_i}=0[/spoiler] is true.

One point I'm not sure about is that I used the equation to work with, but that assumes the d's exists, which assumes that the problem itself has a solution. Eighter I'm missing something, or this has to be done geometrically and constructive. It sounds like an induction problem, but I pretty much hate the word *convex*.

>>4469037
I don't really understand the question, but the (a_i,b_i) are the vertices points, i.e. these expressions are arrows from (0,0) to the corners of the polygon. You can visualize them as a bunch of lines from the origin to the corner, such that you get a star.
The claim is that if you multiply each by the right numbers (the x^n·y^m expressions, which are quite restriced, only two parameters!), i.e. if you line up lines, then you get back to the origin.

-The four momentum has to be summed up just as you expect.
-You can compute all conserved quantities from the Lagrangian of the system, i.e. if there is something rotating than it's because of a potential and you plug that into L.
-The question doesn't have to be treated within general relativity.

Why introduce ghosts in an abelian quantum field theory?