Re: [math-fun] Differential geometry in *finite* fields!
Be careful! Simon Peyton Jones is a friend of mine! The problem with a number of these folks is that they're trying to reduce *too much* of math to a single idea or representation. While it's important to see how far these ideas can be pushed, the true beauty of math IMHO is the ability to look at the same thing from many different angles & many different representations. For example, I'm so upset with the usual discussions of Galois Theory, that I'm contemplating writing a small paper about representing all polynomials by means of square matrices -- i.e., p(x) is represented by a matrix M, such that |M-x*I|=p(x). Most people cringe at using a representation that uses O(n^2) elements for O(n) data, but since we're mathematicians, insight is more important than efficiency, nicht wahr? Perhaps such a paper has already been written; please let me know if you're aware of such a paper. At 05:43 AM 8/3/2017, Fred Lunnon wrote:
Wildberger does himself no favours with the professional community, because he takes no interest in situating his self-invented methods in the broader scheme of established mathematics. I have a good deal of sympathy with his attitude, as it happens; but there does come a point where one must down tools and make an effort to get connected, or else risk wasting further time and temper re-inventing the wheel.
My impression of "Divine Proportions" for example was that it went some of the way towards rediscovering Clifford algebra; but given how that has failed to gain acceptance over the past 144 years, maybe a fresh approach might be no bad thing!
His exuberance and blackboard technique in the video were wondrous to behold. [ I was reminded of another supercharged eccentric, Simon Peyton Jones on Haskell programming --- https://www.youtube.com/watch?v=6COvD8oynmI --- and several more ... ]
WFL
* Henry Baker <hbaker1@pipeline.com> [Aug 03. 2017 18:07]:
Be careful! Simon Peyton Jones is a friend of mine!
The problem with a number of these folks is that they're trying to reduce *too much* of math to a single idea or representation. While it's important to see how far these ideas can be pushed, the true beauty of math IMHO is the ability to look at the same thing from many different angles & many different representations.
For example, I'm so upset with the usual discussions of Galois Theory, that I'm contemplating writing a small paper about representing all polynomials by means of square matrices -- i.e., p(x) is represented by a matrix M, such that |M-x*I|=p(x).
M is the companion matrix of p. The action of M (on another matrix) is a shift of all columns, except for the last where the shift is modulo p. So computing mod p is just a more compact way of working with M. (Warning: I didn't say whether I multiply from left or right, neither which from of companion matrix I use.) I was pretty excited when I observed it, only to much later find out that Richard Brent had mentioned in passing in "On the periods of generalized Fibonacci recurrences", Mathematics of Computation, vol.~63, no.~207, pp.~389-401, (July-1994). He had to go through his own papers to spot the (half)sentence, it is well-hidden. Of course all matrices equivalent to M should do the job as well. Not sure whether some selection of conjugation gives something nice. A book about matrix computations over finite fields is IMO still sorely missing in the market. Btw., about an earlier message of yours concerning diagonalization over a finite field: note that this involves working not over the base field, but the extension field. This may or may not affect whatever you had in mind with that. Best regards, jj
Most people cringe at using a representation that uses O(n^2) elements for O(n) data, but since we're mathematicians, insight is more important than efficiency, nicht wahr?
Perhaps such a paper has already been written; please let me know if you're aware of such a paper.
[...]
Re extensions of extension fields: Think block matrices. Much of the time, a companion matrix is perhaps the *worst* choice for representing a polynomial. Its only raison d'etre is to show the link with the standard monomial basis. For example, for polynomials with all real roots, a symmetric matrix might well be the most convenient choice. At 09:37 AM 8/3/2017, Joerg Arndt wrote:
* Henry Baker <hbaker1@pipeline.com> [Aug 03. 2017 18:07]:
Be careful! Simon Peyton Jones is a friend of mine!
The problem with a number of these folks is that they're trying to reduce *too much* of math to a single idea or representation. While it's important to see how far these ideas can be pushed, the true beauty of math IMHO is the ability to look at the same thing from many different angles & many different representations.
For example, I'm so upset with the usual discussions of Galois Theory, that I'm contemplating writing a small paper about representing all polynomials by means of square matrices -- i.e., p(x) is represented by a matrix M, such that |M-x*I|=p(x).
M is the companion matrix of p. The action of M (on another matrix) is a shift of all columns, except for the last where the shift is modulo p. So computing mod p is just a more compact way of working with M. (Warning: I didn't say whether I multiply from left or right, neither which from of companion matrix I use.)
I was pretty excited when I observed it, only to much later find out that Richard Brent had mentioned in passing in "On the periods of generalized Fibonacci recurrences", Mathematics of Computation, vol.~63, no.~207, pp.~389-401, (July-1994). He had to go through his own papers to spot the (half)sentence, it is well-hidden.
Of course all matrices equivalent to M should do the job as well. Not sure whether some selection of conjugation gives something nice.
A book about matrix computations over finite fields is IMO still sorely missing in the market.
Btw., about an earlier message of yours concerning diagonalization over a finite field: note that this involves working not over the base field, but the extension field. This may or may not affect whatever you had in mind with that.
Best regards, jj
Most people cringe at using a representation that uses O(n^2) elements for O(n) data, but since we're mathematicians, insight is more important than efficiency, nicht wahr?
Perhaps such a paper has already been written; please let me know if you're aware of such a paper.
[...]
participants (2)
-
Henry Baker -
Joerg Arndt