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#1 2007-10-01 11:04:16

Registered: 2005-12-04
Posts: 3,791

Linearly algebraic combinatorics...

...and yes, I made that name up (or so I think).  Anyways, here is the problem: 

S is finite set of polynomials generated by unknowns x and y.  For any positive integer n, Omega_n(S) is the collection of all polynomials formed by multiplying elements of S at most n times.  For example, if S = {x, y}, then Omega_2(S) = {x, y, x^2, xy, y^2}.  Also, let d_n(S) be the number of linear independent polynomials.

Find the least upper bound over all such sets of:

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."


#2 2007-10-24 07:41:43

Registered: 2007-07-16
Posts: 301

Re: Linearly algebraic combinatorics...

Woah, cool. Care to explain what the answer is? I have the feeling it'll be a bit over my head...

Most of that made sense to me (I had to read through it a few times) but what does that line over the limit mean? That's the first time I've seen that.

(Just a comprehension check. Omega_3(S) = {x, y, x², y², x³, y³, xy, x²y, xy²})

There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction.


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