Magma Computation needed !

Jürgen Böhm · science forum beginner Joined: 10 Jun 2005 Posts: 15

Hello,

maybe someone with access to MAGMA can help me and try to execute the
following, probably quite expensive computation on his computer:

Z:=IntegerRing();
L<x, ac, pa, pb, pc, aa, bb, qa, qb, pd, qc, qd, ab, ad, ba, bc, bd, ra
, rb, rc, sb, sc, sd, ub, uc, ud, wa, wb, wc, A,
B>:=PolynomialRing(Z,31,"elim",29);
I:=ideal<L| pd*pb^2-pa*pc^2, qd*pb^2+2*pd*qb*pb-qa*pc^2-2*pa*qc*pc,
aa*pa+ba*qa-1
, ab*pb+bb*qb-1, ac*pc+bc*qc-1, ad*pd+bd*qd-1, ra*pa+sb*pb-1,
rb*pb+sc*pc-x^3-
A*x-B, rc*pc+sd*pd-1, wa*pa+ra*qa+ub*pb+sb*qb,
wb*pb+rb*qb+uc*pc+sc*qc-3*x^2-A
, wc*pc+rc*qc+ud*pd+sd*qd>;

Groebner(I);

I tried it with the MAGMA online calculator
http://modular.fas.harvard.edu/cgi-bin/calc/calc.py but it consumed too
much time and memory (limited there to 20 seconds). I would be very much
interested in the result, actually the question is whether n1 *
(4*A^3+27*B^2)^n2 with integers n1 and n2 appears in the Groebner-base.

Thanks in advance

Jürgen Böhm

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"At a time when so many scholars in the world are calculating, is it not
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Roman Pearce · science forum beginner Joined: 03 May 2005 Posts: 37

I'm getting down to a 120480 by 333717 matrix in the F4 algorithm.
Now, since this is a matrix over Z I'm going to guess that you're
doomed. However, I'll let it run for a few days (2.5 GHz PPC G5).

Roman

Roman Pearce · science forum beginner Joined: 03 May 2005 Posts: 37

Sure enough I ran out of memory (4GB). You may be able to do this with
an 8GB machine. Does it have to be over the ring of integers, or did
you want rational number coefficients ?

Jürgen Böhm · science forum beginner Joined: 10 Jun 2005 Posts: 15

Roman Pearce wrote:

Roman Pearce · science forum beginner Joined: 03 May 2005 Posts: 37

Well, I'm a graduate student so things like computer time and Magma
licenses are much cheaper Smile

I used an Apple PPC machine. You can
buy PPC machines from IBM as well, but they are generally quite
expensive (with server-grade hardware, etc).

As for your problem, computing Groebner bases over a finite field is
much faster than either the rational numbers or over Z, especially in
Magma. Will it help you to do this problem over GF(p^n) ?

Roman

Jürgen Böhm · science forum beginner Joined: 10 Jun 2005 Posts: 15

Roman Pearce wrote:

Google