Le 08/04/2020 à 01:03, Nils Bruin a écrit :
On Tuesday, April 7, 2020 at 3:10:00 PM UTC-7, David Roe wrote:
For matrices over Q there's
sage.matrix.misc.matrix_rational_echelon_form_multimodular, which is the
default for matrices with more than 25 rows/columns. It should be possible
to adapt this to number fields.
In particular, by Weil restriction you can consider your matrix as a
matrix over Q. You'll get a subspace of Q^(20*30), but the multimodular
methods implemented for that should be able to get you good rank
information fairly quickly: the rank over K should just be that rank
divided by 20 (if it doesn't come out as a multiple of 20, something went
wrong).
Hi Nils,
How do you make the Weil restriction happen? Is this the following
Let K = Q[a] be of degree d. For each row v of the matrix, make
d new rows for the new matrix with a, a*v, a^2*v, ..., a^(d-1)*v
seen as elements of Q^(d * ncols).
If so, I will give it a try. One inconvenient with this approach is
that you are duplicating a lot the information. A priori, the data
fits into QQ^(nrows * ncols * d) but the above procedure constructs
a matrix over QQ^(nrows * ncols * d * d). As you said, we get d times
the rank in the end.
Best
Vincent
PS: In PARI there is a bunch of ZabM_* functions that works over
cyclotomic fields (though not available in GP). We also have
something in Sage specialized for cyclotomic field. I am
investigating on this side as well (my fields are subfields
of cyclotomic fields that gives an extension at most 4).
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