## Elliptic Curves Are Curves of Constant Influence - Doctorow

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• From: Osher Doctorow, Ph.D. osher@ix.netcom.com, Sunday Feb. 4, 2001 6:35AM I have been developing the theory of proximity functions p1(x,y) = 1 + y - x and
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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
and its n-dimensional generalization pn(x1, x2, ..., xn; y) = 1 + y - Sn/n
where Sn = x1 + x2 + ... + xn and 0 < = y < = xi < = 1 for i = 1, 2, ..., n.
See geometry-research@... for more details, although I will
mention that p1 and pn change distance-functions or metrics of usual type on
[0, infinity) to partial inverses on [0, 1] and are maximum where, e.g.,
Euclidean distance-function is minimum and vice versa, subject to the
constraints 0 < = y < = xi < = 1. They satisfy the triangle inequality,
nonnegativity (in fact, they are between 0 and 1), and "reflexivity by
default" since the only pair for which (x,y) and (y,x) both exist subject to
the constraints is (0,0). Those familiar with my earlier contributions to
primenumbers or primes-L will recognize in p1 the probable influence
P(A-->B) = 1 + P(AB) - P(A) for y = P(AB) and x = P(A), and of course from
elementary probability and set theory it follows that 0 < = P(AB) < = P(A) <
= 1 where AB is the intersection of A and B and P(A) is the probability of
A. I will only discuss p1 in what follows.

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).
Define a new ordering for arbitrary polynomials Si with real coefficients
up to symmetric terms in x and y by Si > = Sj iff Si and Sj are monic with
Si having degree at least as great as the degree of Sj up to symmetric terms
in x and y, although > would be a slightly better notation since = is
usually reserved for the usual equality. 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).

Osher Doctorow
Doctorow Consultants, West Los Angeles College, Ventura College, etc.
• 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 2 of 2 , Feb 4, 2001
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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.