On 5/2/07, mabshoff <Michael.Abshoff@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx> wrote:
>
> > (2) Singular *is* capable of handling the ring Integers(3**6)['x,y'].
> > Check it out (ok, I use 3**5 below):
> >
> > sage: r = singular.ring(3^5, '(x,y)', 'dp')
> > sage: r
> >
> > // characteristic : 241
> > // number of vars : 2
> > // block 1 : ordering dp
> > // : names x y
> > // block 2 : ordering C
> > sage: singular('(x+y)^10')
>
> This gives you F_241 - as fas as I know singular returns the ring F_p
> with p the next smallest prime if n is composite. Here instead of
> F_243 you get F_241. I do not know if you can actually get F_n with n
> composite in singular, but Martin probably knows.
That's just evil. Evil. I need to change the SAGE singular.ring
command to raise an error in such cases.
> > x^10+10*x^9*y+45*x^8*y^2+120*x^7*y^3-31*x^6*y^4+11*x^5*y^5-31*x^4*y^6+120*x^3*y^7+45*x^2*y^8+10*x*y^9+y^10
> > sage: R.<x,y> = Integers(3^5)[x,y]
> > sage: (x+y)^10
> > y^10 + 10*x*y^9 + 45*x^2*y^8 + 120*x^3*y^7 + 210*x^4*y^6 + 9*x^5*y^5 +
> > 210*x^6*y^4 + 120*x^7*y^3 + 45*x^8*y^2 + 10*x^9*y + x^10
> >
> > So singular seems to be capable of *arithmetic* modulo n for any n.
> > It just doesn't do
> > Groebner basis modulo n for n composite.
> >
>
> The two results from above are different, -31*x^4*y^6 is equal to
> 210*x^4*y^6 in F_241, not F_243.
>
Thanks for pointing this out.
> <shameless plug>CoCoALib can do multivariate polynomial arithmetic in
> Z_n, the bindings Martin and I wrote (well mostly Martin) only do QQ
> at the moment, but Z_n is just a copy and paste job once the bindings
> have been fleshed out.</shameless plug>
Awesome! I'm really really not interested in reinventing the wheel
for basic multivariate polynomial arithmetic. This will be an
excellent reason to include Cocoa in the core of SAGE.
> I have been rather busy the last month, so my time on sage was mostly
> spend on installers and testing. Hopefully once 2.5 is out I can spend
> more time on CoCoALib bindings.
Thanks for responding to the email. I'm pretty amazed the Singular
would dare to just take the previous prime without an error...
--
William Stein
Associate Professor of Mathematics
University of Washington
http://www.williamstein.org
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