Heights of Ideals

[Mike]

Under the ring declaration (ring
R=integer,(a1,a2,a3,a4,b1,b2,b3,b4),dp;), I was unable to compute the
height of the following ideal:

> ideal I=3*a2*b1^2+3*a1^2*b2+6*a3*b1*b2+3*a4*b2^2,6*a1^2*b1^2+4*a3*b1^3+24*a1*a2*b1*b2+12*a4*b1^2*b2+6*a2^2*b2^2+12*a1*a3*b2^2,20*a1*a2*b1^3+5*a4*b1^4+20*a1^3*b1*b2+30*a2^2*b1^2*b2+60*a1*a3*b1^2*b2+30*a1^2*a2*b2^2+60*a2*a3*b1*b2^2+60*a1*a4*b1*b2^2+10*a3^2*b2^3+20*a2*a4*b2^3,20*a1^3*b1^3+15*a2^2*b1^4+30*a1*a3*b1^4+180*a1^2*a2*b1^2*b2+120*a2*a3*b1^3*b2+120*a1*a4*b1^3*b2+15*a1^4*b2^2+180*a1*a2^2*b1*b2^2+180*a1^2*a3*b1*b2^2+90*a3^2*b1^2*b2^2+180*a2*a4*b1^2*b2^2+20*a2^3*b2^3+120*a1*a2*a3*b2^3+60*a1^2*a4*b2^3+120*a3*a4*b1*b2^3+15*a4^2*b2^4
. ;

> I;
I[1]=3*a2*b1^2+3*a1^2*b2+6*a3*b1*b2+3*a4*b2^2
I[2]=6*a1^2*b1^2+4*a3*b1^3+24*a1*a2*b1*b2+12*a4*b1^2*b2+6*a2^2*b2^2+12*a1*a3*b2^2
I[3]=20*a1*a2*b1^3+5*a4*b1^4+20*a1^3*b1*b2+30*a2^2*b1^2*b2+60*a1*a3*b1^2*b2+30*a1^2*a2*b2^2+60*a2*a3*b1*b2^2+60*a1*a4*b1*b2^2+10*a3^2*b2^3+20*a2*a4*b2^3
I[4]=20*a1^3*b1^3+15*a2^2*b1^4+30*a1*a3*b1^4+180*a1^2*a2*b1^2*b2+120*a2*a3*b1^3*b2+120*a1*a4*b1^3*b2+15*a1^4*b2^2+180*a1*a2^2*b1*b2^2+180*a1^2*a3*b1*b2^2+90*a3^2*b1^2*b2^2+180*a2*a4*b1^2*b2^2+20*a2^3*b2^3+120*a1*a2*a3*b2^3+60*a1^2*a4*b2^3+120*a3*a4*b1*b2^3+15*a4^2*b2^4

> heightZ(I);

Also, while working under Sage, I was able to compute heights of
ideals, but unable to compute the above height as well in SAGE.

[bogo...@math.uni-hannover.de]

Zitat von Mike <calimi...@gmail.com>:

Hi Mike,
I tried it with our algorithm "myheight(I)" in Singular and the result was 2.
It took about 12 minutes....
Regards,
Xenia

[Michael Garcia]

Hi Xenia,

Thank you for the help you provided earlier.  Regarding the algorithm you used that took 12 minutes, is that what I need to use? I am having trouble with my computations in Sage, so should I use Singular directly?

If that's the case, when I open in the terminal, this is what's displayed:

Last login: Tue Aug  9 20:32:23 on ttys003
/Applications/Sage-4.7-OSX-64bit-10.6.app/Contents/Resources/sage/sage --singular; exit
Michael-Garcias-MacBook-Pro:~ Michael$ /Applications/Sage-4.7-OSX-64bit-10.6.app/Contents/Resources/sage/sage --singular; exit
Detected SAGE64 flag
Building Sage on OS X in 64-bit mode
                    SINGULAR                             /  Development
A Computer Algebra System for Polynomial Computations   /   version 3-1-1
                                                      0<
    by: G.-M. Greuel, G. Pfister, H. Schoenemann        \   Feb 2010
FB Mathematik der Universitaet, D-67653 Kaiserslautern    \
>

So I think it's being opened within Sage, which is not what I want, right?  My end goal is to be able to compute this ideal with integer polynomials in a multivariate ring over the rationals.  I shouldn't have any problems regarding run time, correct?

Hopefully somebody can help out.

Thank you for your time,

Michael

[bogo...@math.uni-hannover.de]

Hello Michael,
if you use Singular directly and declare a ring over the integers, as
I did, then the following is diplayed:

SINGULAR /
A Computer Algebra System for Polynomial Computations / version 3-1-2
0<
by: W. Decker, G.-M. Greuel, G. Pfister, H. Schoenemann \ Oct 2010

FB Mathematik der Universitaet, D-67653 Kaiserslautern \

> ring r=integer,(x,y,z),dp;
// ** You are using coefficient rings which are not fields.
// ** Please note that only limited functionality is available
// ** for these coefficients.
// **
// ** The following commands are meant to work:
// ** - basic polynomial arithmetic
// ** - std
// ** - syz
// ** - lift
// ** - reduce
>

The dimension of Z[x_1,...,x_n] is n+1 and if you have an ideal of
dimension k over the integers, the ideal generated by the same
polynomials over the rationals will be of dimension k-1. So the height
will be n+1-k over the integers and over the rationals.

So if you want to compute the height of an ideal I over the rationals,
just use Singular, declare the ring r=0,(x_1,..,x_n),dp(for example);
and use the function dim(I); . The height(I) will be n-dim(I).
I hope that answers your question.
We often work over tghe integers and have some experimental algorithms
for that.
So good luck!
Xenia

Zitat von "Michael Garcia" <calimi...@gmail.com>: