On Tuesday, April 7, 2020 at 11:42:41 PM UTC-7, vdelecroix wrote: > > 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). > > Exactly. If you're worried about space usage and are happy to only be morally sure of your answer, you can reduce the matrix modulo a whole bunch of primes and compute the rank there. Over finite fields (also non-primitive ones), I'm sure the rank computation shouldn't be a huge hurdle.
The rank bound in reduction will be a lower bound on the rank. If you compute enough primes, then it will also be an upper bound: you can bound the height of the rxr minors, so if you're finding that they're all zero modulo enough primes that the smallest non-zero field element that reduces to 0 modulo all the primes you have selected exceeds that bound, then you've proven the rank isn't larger than what you've found. That's basically what the multimodular algorithms would do, so it's probably way less work to convert to a matrix over Q. > 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). > -- You received this message because you are subscribed to the Google Groups "sage-devel" group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-devel+unsubscr...@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/sage-devel/e084880a-7275-4ab1-b87e-164cea418e57%40googlegroups.com.
