The following 2D example describes the problem:
PR ==> PAdicRational(11)
SQ ==> SQMATRIX(2, PR)
LS ==> List SQ
M : SQ := [[10, 7], [3, 8]]
┌10 7┐
(4) │ │
└3 8┘
N : SQ := [[9, 2], [5, 1]]
┌9 2┐
(5) │ │
└5 1┘
S : LS := [M, N]
┌10 7┐ ┌9 2┐
(6) [│ │, │ │]
└3 8┘ └5 1┘
AP := ALGSC(PR, 2, ['A, 'B], S)
(7)
AlgebraGivenByStructuralConstants(PAdicRational(11),2,[A,B],[[[10,7],[3,8]],[
[9,2],[5,1]]])
V := basis()$AP
(8) [A, B]
(A, B) := (V.1, V.2)
(9) B
)clear p M N S V
(10) -> (U, V, W) : AP
(11) -> U := 9*A + 10*B
(11) 10 B + 9 A
(12) -> V := 8*A + 7*B
(12) 7 B + 8 A
(13) -> W := U*V
2
2 3
(13) (1 + 3 11 + 10 11 )B + (3 + 2 11 + 5 11 + 11 )A
(14) ->
All scalar coefs are padic integers and an algebra over the ring
of padic integers would've been sufficient. However, ALGSC requires
the Ring to be a Field, so, PAdicRational was chosen, instead.
The problem now is to rewrite the product W into the following form
or into something equivalent:
2 3
(B + 3A) + (3B + 2A)11 + (10B + 5A)11 + A 11
where the powers of the prime (11, in this case) are factored out.
SWA
On Wednesday, April 19, 2023 at 7:06:30 PM UTC-5 Waldek Hebisch wrote:
> On Tue, Apr 18, 2023 at 07:48:48AM -0700, Sid Andal wrote:
> >
> > The following two vectors
> >
> > U := [2, 3, 4]
> > V := [7, 8, 9]
> >
> > are in a 3-Dim Alg over the PAdicRational(13).
> >
> > Is there a way to rewrite their product
> >
> > 2 2 3
> > W := [6 + 13 + 7 13 , 1 + 9 13 + 7 13 , 8 13 + 7 13 ]
> >
> > in a different form where the powers of 13 are factored out of the
> brackets
> > per the following?
> >
> > 2
> > 3
> > W := [6, 1, 0] + [1, 9, 8] 13 + [7, 7, 0] 13 + [0, 0, 7] 13
> >
>
> You did not write what your really want. Do you want internal
> structure to be as above? If yes, than you need new domain
> for vectors (or whatever aggregate you use). Such domain would
> have to do similar things to what PAdicRational is doing, but
> using stream of vectors (or more general aggregates).
>
> If all what you want is printing, then you should be able to
> produce printouts using digits extraction (say via 'approximate').
>
> --
> Waldek Hebisch
>
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