Re: ellisoncurve() and ellpointtoz() [2]

[Vasily Golyshev <golyshev@mccme.ru>]

On Wed, 16 Oct 2002, Karim BELABAS wrote:

> On Mon, 14 Oct 2002, Vasily Golyshev wrote:
> > here is an irregularity in ellisoncurve():
> >
> > ? C=ellinit([0,0,0,0,1])
> > %1 = [0, 0, 0, 0, 1, 0, 0, 4, 0, 0, -864, -432, 0,
> > [-1.000000000000000000000000000, 0.5000000000000000000000000000 -
> > 0.8660254037844386467637231707*I, 0.5000000000000000000000000000 +
> > 0.8660254037844386467637231707*I]~, 4.206546315976362783525057237,
> > -2.103273157988181391762528618 + 1.214325323943790805909970844*I,
> > -1.293554779614895267476757512 + 1.44309482 E-29*I,
> > 0.6467773898074476337383787562 - 1.120251300333280219655206320*I,
> > 5.108115717832556535122194506]
> >
> >
> > ? ellisoncurve(C,[Mod(t,t^2-t+1),0])
> > %2 = 1
> > ? ellisoncurve(C,[1/2-sqrt(3)/2*I,0])
> > %3 = 0
> >
> > Now, what I actually need for my purposes is to find
> > (up to a sign) ellpointtoz() of a point, given only its abscissa,
> > but it appears that I have to do
> >
> >  ellpointtoz(C,[p[1],ellordinate(C,p[1])[1]])
> >
> > first, which fails in this case.
>
> [Cc-ed to pari-dev]
>
> For non-exact input, ellisoncurve() is quite kludgy (it needs to decide
> whether some floating point coordinates are non-zero).
>
> There was a bug in the current kludge which caused it to err when the
> y-coordinate was exactly 0. I've complicated it further in order to fix this
> specific problem.
>
> But this is really not a proper fix. In fact, the only reliable fix I see is
> to decline to check that the point is on the curve when some of the data is
> inexact. Currently, only ellpointtoz and ellheight perform this check. I am
> not sure why, or whether it should not be cancelled altogether (possibly
> unless debugging level is high enough).
>
> Cheers,
>
>     Karim.
>

It seems to me that ellheight is something that you would only want
to apply to exact data. Ellpointtoz, on the other hand, will be safer with
a possibility of a check. A compromise could be an ellpointtoz with a
'float' flag, returning the result and also the abs(RHS-LHS) of the
defining equation, for the user to decide for himself whether to consider
the result valid.

Vasily.