Algebraic independence is driving me around the bend. In the matroid paper I'm reading ("Matroids from Discrete Geometry" by Walter Whiteley), it states "if the coordinates of a configuration are algebraically independent reals then the configuration will be generic."
"Generic" is an important concept in rigidity matroids. I understand generic. I'd like to understand "algebraically independent reals" so that I can, say, generate a set of generic coordinates at will. Most embeddings will be generic, but not all are. When you test the rank of the rigidity matrix and it is not full rank, that means that either there are dependencies, or the coordinates are not generic. I'd like to know which. I'm not using this method in my software, but it's always good to understand alternate methods... they're particularly good at verifying results. ;)
I've googled "algebraically independent", and found descriptions at sites like MathWorld and PlanetMath, but this bear of very little brain is having trouble parsing them. I need some good, concrete examples. Anyone have any suggestions?
On the up side, after the lunchtime group meeting, I'm now about 70% done the chicken socks. ;) (I didn't do any knitting or crochet last night because I wanted to finish the Yarn Harlot's book, so I could give it back at SNB tonight.)
2 comments:
Algebraic independence is a concept that applies to real numbers (e.g. pi is transcendental, meaning it's algebraically independent from the rational numbers). Generating coordinates is a concept that applies to floating point numbers. The two are not the same thing and it is probably a mistake to try to pretend that they are (e.g., for any set of floating point numbers you give me, I can find a set of algebraically independent numbers that are within the floating point precision of the ones you gave me).
If you want to be sure you're getting this right, I imagine you need to replace the numbers in your rigidity matrix with polynomials, where the variables of your polynomials are the algebraically independent values you need. I.e., leave the coordinates of the points as the symbolic values (x0,y0), (x1,y1), ... instead of filling them in with numbers, and work out the rank of the matrix using symbolic arithmetic on polynomials.
Copy of reponse sent to DE:
Thanks! That helps a bit.
So all transcendental numbers are algebraically independent from the reals?
Do you have any idea if the coordinates are supposed to be algebraically independent from each other, or just from the reals? I suppose that was a silly question... since transcendental numbers are themselves reals, they'll all be independent from each other by definition. Actually, that gets me confused over another wikipedia entry, since
http://en.wikipedia.org/wiki/Algebraically_independent
says that it's not known if the set {\pi, e} is algebraically independent over Q.
I'm not that concerned with floating point, since I'm really trying for a mathematical understanding, rather than adding this to my code.
Just before Walter states that algebraically independent coordinates will be generic, he discusses how to find the rank by doing as (I think) you suggest: he replaces the coordinates by indeterminants, and finds the rank by finding maximal non-zero minors. That part I understand.
Thanks!
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