On 9/21/07, John Cremona <[EMAIL PROTECTED]> wrote:
>
> It *is* a ternary quadratic form once you homogenize with a 3rd variable z.
>
> Finding rational points on plane conics (which is what this is) has
> advanced substantially in the last few years. My paper with Rusin
> (Mathematics of Computation, 72 (2003), no. 243, pages 1417-1441.)
> works well for diaginal ones and is behind Magma's first
> implementations for RationalPoint(Conic()); a different method by
> Denis Simon is better for non-diagonal ones and is (I believe) what
> Magma uses.
>
> My method is implemented in the C++ code which is already in Sage in
> the mwrank package, so all tat would be needed would be to write the
> appropriate wrappers!
I've made this trac ticket #727:
http://trac.sagemath.org/sage_trac/ticket/727
Volunteer(s) needed!
-- William
> John
>
> On 9/21/07, David Stahl <[EMAIL PROTECTED]> wrote:
> >
> > Hi Utpal,
> >
> > Does the Hasse-Minkowski theorem apply for a non-quadratic form like
> > mine?
> >
> > David
> >
> > On Sep 20, 2:34 pm, Utpal Sarkar <[EMAIL PROTECTED]> wrote:
> > > There is not always a solution. Whether or not there is a solution is
> > > the contents of the Hasse-Minkowski theorem. I couldn't find a
> > > function in sage that immediately tells you whether there is a
> > > rational solution. There is a function that tells you whether there is
> > > a local solution at a prime p, namely hilbert_symbol(-N, d, p) (this
> > > is 1 when there is a solution, otherwise -1), and the Hasse-Minkowski
> > > theorem actually states that there is a global (rational) solution if
> > > and only if there is a local solution at every prime p including
> > > infinity (in sage you have to pass p = -1). In fact this only has to
> > > be checked for primes that divide N or d, for 2 and for infinity.
> > >
> > > In sage you could write a function like this, in one line if you use
> > > some fancy python constructs (using the N and d as in your equation
> > > (2), check just in case I made a mistake):
> > > def has_rational_solution(N,d):
> > > return reduce(lambda P,Q: P and Q, [prod([hilbert_symbol(a,b,p)
> > > for a in [-N.numerator(), N.denominator()] for b in [d.numerator(),
> > > d.denominator()]]) == 1 for p in prime_divisors(2*N*d) + [-1]])
> > >
> > > If you have magma installed (accessible from sage in that case), then
> > > this function will actually give you a rational point (in homogeneous
> > > coordinates) if it exists:
> > > f := func<N,d| HasRationalPoint(Conic(P2, P2.1^2 - d*P2.2^2 +
> > > N*P2.3^2))> where P2 is ProjectiveSpace(Rationals(),2);
> > >
> > > Hope you find this useful.
> > > Greetings,
> > >
> > > Utpal
> > >
> > > On Sep 20, 9:40 pm, David Stahl <[EMAIL PROTECTED]> wrote:
> > >
> > >
> > >
> > > > I have a non-SAGE question and am hoping someone can point me to a
> > > > source that discusses the solution. I am trying to find a rational
> > > > solution for x and y to the equation:
> > >
> > > > Ax^2 + Bxy + Cy^2 + Dx + Ey + F =0 (1)
> > >
> > > > where the coefficients are rational. This can be transformed to:
> > >
> > > > xprm^2 - d*yprm^2 + N = 0 (2)
> > >
> > > > There are alot of websites that talk about finding integer solutions
> > > > to these equations with integer coefficients. I do not think an
> > > > integer solution always exists when the coefficients of (2) are
> > > > rational but I do think a rational solution always does exist and I am
> > > > perfectly happy with a rational solution. Any guidance would be
> > > > appreciated. Thank you.
> > >
> > > > David- Hide quoted text -
> > >
> > > - Show quoted text -
> >
> >
> > >
> >
>
>
> --
> John Cremona
>
> >
>
--
William Stein
Associate Professor of Mathematics
University of Washington
http://wstein.org
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