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 - --~--~---------~--~----~------------~-------~--~----~ To post to this group, send email to sage-support@googlegroups.com To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/sage-support URLs: http://sage.math.washington.edu/sage/ and http://sage.scipy.org/sage/ -~----------~----~----~----~------~----~------~--~---
