fivemack's profile - activity

Just write M.LLL() instead of M.lll()

I think you want the .lift() method:

sage: P.<x,y,z,t>=PolynomialRing(QQ,4) 
sage: C1= 17*x^2 + 7*y^2 - 26*y*z + 7*z^2
sage: C2= 13*y^2 - 7*y*z + 13*z^2 - 51*t^2
sage: f = x^4+y^4+z^4-18*t^4
sage: 
sage: f in ideal(C1,C2)
True
sage: 
sage: CI = ideal(C1, C2)
sage: coords = f.lift(CI)
sage: coords
[1/17*x^2 + 1/13*y*z - 21/221*t^2, -7/221*x^2 + 1/13*y^2 + 1/13*z^2 + 6/17*t^2]
sage: rebuilt = sum(coord*base for coord, base in zip(coords, [C1, C2]))
sage: rebuilt
x^4 + y^4 + z^4 - 18*t^4
sage: f == rebuilt
True

P.<x,y,z,t>=PolynomialRing(QQ,4) 
C1= 17*x^2 + 7*y^2 - 26*y*z + 7*z^2
C2= 13*y^2 - 7*y*z + 13*z^2 - 51*t^2
x^4+y^4+z^4-18*t^4 in ideal(C1,C2)

So there exist \alpha, \beta in P with <latex>\alpha C1 + \beta C2 = x^4+y^4+z^4-18*t^4</latex>

What's the command to find \alpha and \beta explicitly ?

Ah yes, thanks!

I'm trying to find elliptic curves lying on a quartic surface; there are probably clever ways to do this, but at the moment I am finding lots of quadrics through an integer point on the surface, then iterating over pairs of them and checking

x^4+y^4+z^4-67*t^4 in ideal(f1,f2)

This leaks about 2MB memory per second, presumably because the Groebner bases for the ideals are being cached; is there a way to tell sage not to cache them?

I am looking for quadratic forms with a given point - so I want short integer vectors which are perpendicular to

v=[X*X,X*Y,X*Z,X*T,Y*Y,Y*Z,Y*T,Z*Z,Z*T,T*T]

for (in this case) [X,Y,Z,T]=[4423,7583,8765,3459]

This sounds like a problem with LLL written all over it, so I do

  L=[[Y,-X,0,0,0,0,0,0,0,0],[0,Z,-Y,0,0,0,0,0,0,0],[0,0,T,-Z,0,0,0,0,0,0],[0,0,0,Y*Y,-X*T,0,0,0,0,0],[0,0,0,0,Z,-Y,0,0,0,0], 
  [0,0,0,0,0,T,-Z,0,0,0],[0,0,0,0,0,0,Z*Z,-Y*T,0,0],[0,0,0,0,0,0,0,T,-Z,0],[0,0,0,0,0,0,0,0,T,-Z]]

  M=matrix(L)

  M.lll()

but this gives an error message AttributeError: 'sage.matrix.matrix_integer_dense.Matrix_integer_dense' object has no attribute 'lll'