Re:[PrimeNumbers] Elliptic Curves Are Curves of Constant Influence - Doctorow
- In a message dated 04/02/2001 15:13:53 GMT Standard Time, osher@...
> From: Osher Doctorow, Ph.D. osher@..., Sunday Feb. 4, 2001 6:35AM[snip]
> I have been developing the theory of proximity functions p1(x,y) = 1 + y - x
> For the elliptic curve y^2 + c1xy + c2y = x^3 + c3x^2 + c4x + c5 where ci[I'm afraid absestos underwear is needed at this point...]
> for i = 1 to 5 are real numbers, the left hand side is a polynomial in y up
> to symmetric terms in x and y (namely c1xy), which will be denoted by F(y),
> and the right hand side is a polynomial in x which will be denoted by G(x).
> For the elliptic curve, certainly
> F(x) < = G(y), and we can add 1 to both sides of the elliptic curve equation
> above and then incorporate the left-hand side 1 into F which now has an
> additional constant term 1 which new polynomial will still be written F, so
> that now we have F(y) = G(x) + 1 or 1 + G(x) = F(y) where = is used in its
> ordinary sense. Since the indeterminate symbols are arbitrary, writing 1 +
> G(y) - F(x) = 0 yields the form 1 + y - x = 0 for y replaced by G(y) and x
> replaced by F(x). Thus, the elliptic curve is a curve of 0 generalized
> influence, and since a constant was incorporated into F, it is correct to
> say that the elliptic curve is a curve of in general constant influence.
> Elliptic curves can be regarded as number theory analogues of geodesics in
> view of this result (in the sense of constant parallel displacement).
So, if I have a curve F(y) = G(x), where F and G are _arbitrary_ monic
and do your replacements on them, I get 1 + F(y) - G(x) = 0,
and thence 1 + y - x = 0,
and can then say my curve is "an analogy to a geodesic."
But, a geodesic is a _special_ kind of curve, namely one determined by
self-parallel displacement (w.r.t a particular affine connection) of the
tangent vector to the curve.
This reductio ad absurdum shows that your observations are without any
Please, also, look back over your article and notice how the notation
between G(y) and F(x), and F(y) and G(x).
How are we supposed to make any sense at all of such a posting?