## Re:[PrimeNumbers] Elliptic Curves Are Curves of Constant Influence - Doctorow

Expand Messages
• In a message dated 04/02/2001 15:13:53 GMT Standard Time, osher@ix.netcom.com ... [snip] ... [I m afraid absestos underwear is needed at this point...] So, if
Message 1 of 2 , Feb 4, 2001
• 0 Attachment
In a message dated 04/02/2001 15:13:53 GMT Standard Time, osher@...
writes:
> From: Osher Doctorow, Ph.D. osher@..., Sunday Feb. 4, 2001 6:35AM
> I have been developing the theory of proximity functions p1(x,y) = 1 + y - x
[snip]
> For the elliptic curve y^2 + c1xy + c2y = x^3 + c3x^2 + c4x + c5 where ci
> 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).
>[snip]
> 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).

[I'm afraid absestos underwear is needed at this point...]

So, if I have a curve F(y) = G(x), where F and G are _arbitrary_ monic
polynomials,
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.