Saturday, May 12, 2007
JSH: Analyzing algebraic residues
Years ago I came up with the idea of subtracting equations out of identities, so like you can consider
x^2 + y^2 + vz^2 = x^2 + y^2 + vz^2
and subtract out x^3 + y^3 = z^3, and analyze what's left—the algebraic residue.
Since this approach is about residues, you use congruences so you'd have
x^2 + y^2 + vz^2 = 0(mod x^2 + y^2 + vz^2)
and when I introduced this technique back in late 1999 and early 2000 on math newsgroups this approach was so new that for a long time I had arguments just over the FORM as some couldn't quite understand how
x^2 + y^2 + vz^2 = 0(mod x^2 + y^2 + vz^2)
was equivalent to
x^2 + y^2 + vz^2 = x^2 + y^2 + vz^2.
So why bother with identities and subtracting out an equation like x^3 + y^3 = z^3?
Because what's left over—the algebraic residue—is constrained by the equation you've subtracted out, so that what is true for the residue must be true for the original and vice versa.
The algebraic residue is a mathematical shadow of the original equation, but there is one key addition—another variable.
So you can analyze the residue by adjusting the free variable, and I used this technique with great success and the mathematical world did not yawn.
It blew up, as I didn't just argue on Usenet but emailed mathematicians at universities and even visited one at Vanderbilt University, and wrote a paper which was published by the now defunct Southwest Journal of Pure and Applied Mathematics aka SWJPAM.
That journal died after publishing a key paper very closely related to this area, when sci.math posters mounted a successful email campaign to break the peer review system—as my paper passed it—so that they could maintain a proof was in error.
Your society already broke, years ago, you may have just not noticed it.
The mathematical world that exists today is not the one that was here before my paper, no matter how successfully mathematicians hide that things have changed.
Why the big deal?
Because this simple idea of subtracting equations out and analyzing the residue revealed problems with previous strongly held ideas so one dead electronic mathematical journal is just the dead fish floating on the water.
One dead math journal not worth the world's news.
x^2 + y^2 + vz^2 = x^2 + y^2 + vz^2
and subtract out x^3 + y^3 = z^3, and analyze what's left—the algebraic residue.
Since this approach is about residues, you use congruences so you'd have
x^2 + y^2 + vz^2 = 0(mod x^2 + y^2 + vz^2)
and when I introduced this technique back in late 1999 and early 2000 on math newsgroups this approach was so new that for a long time I had arguments just over the FORM as some couldn't quite understand how
x^2 + y^2 + vz^2 = 0(mod x^2 + y^2 + vz^2)
was equivalent to
x^2 + y^2 + vz^2 = x^2 + y^2 + vz^2.
So why bother with identities and subtracting out an equation like x^3 + y^3 = z^3?
Because what's left over—the algebraic residue—is constrained by the equation you've subtracted out, so that what is true for the residue must be true for the original and vice versa.
The algebraic residue is a mathematical shadow of the original equation, but there is one key addition—another variable.
So you can analyze the residue by adjusting the free variable, and I used this technique with great success and the mathematical world did not yawn.
It blew up, as I didn't just argue on Usenet but emailed mathematicians at universities and even visited one at Vanderbilt University, and wrote a paper which was published by the now defunct Southwest Journal of Pure and Applied Mathematics aka SWJPAM.
That journal died after publishing a key paper very closely related to this area, when sci.math posters mounted a successful email campaign to break the peer review system—as my paper passed it—so that they could maintain a proof was in error.
Your society already broke, years ago, you may have just not noticed it.
The mathematical world that exists today is not the one that was here before my paper, no matter how successfully mathematicians hide that things have changed.
Why the big deal?
Because this simple idea of subtracting equations out and analyzing the residue revealed problems with previous strongly held ideas so one dead electronic mathematical journal is just the dead fish floating on the water.
One dead math journal not worth the world's news.
# posted by James Harris @ 19:54 
