Here is a speed comparison with Singular on a well known computation (cyclic) and a one less so (meyer).

All computations were made on a Opteron 1.6GHz with 2GB. The java virtual machine is "Java HotSpot(TM) Server VM (build Blackdown-1.4.2-rc1, mixed mode)" with 400MB initial and maximum heap size.

The software versions are:
Singular-2-0-3-ix86-Linux                (2002/02)
j2re-1.4.2-rc1-linux-i586-gcc3.2        (2003/12/18)
meditor1.0                                (2004/02/22)

All times are given in seconds.

1) cyclic, DegreeReverseLexicographic, BigIntegers

        | meditor             | Singular | ratio  |
--------+---------------------+----------+--------+
cyclic4 | 0.018               | 0.000    |        |
cyclic5 | 0.583               | 0.005    | 116.60 |
cyclic6 | 5.010               | 0.323    |  15.51 |
cyclic7 | 1.165e6 (13.5 days) | (*)      |        |
--------+---------------------+----------+--------+

(*) the program stops by itself at 1GB

2) cyclic, DegreeReverseLexicographic, modulo p (p=32003)

        | meditor             | Singular | ratio  |
--------+---------------------+----------+--------+
cyclic4 |   0.018             |   0.000  |        |
cyclic5 |   0.392             |   0.003  | 130.66 |
cyclic6 |   1.483             |   0.042  |  35.30 |
cyclic7 | 128.940             |   4.813  |  26.78 |
cyclic8 |  15.480e3 (4h18min) | 142.267  | 108.80 |
--------+---------------------+----------+--------+

3) meyer, Lexicographic, (boolean)

       | meditor | Singular | ratio |
-------+---------+----------+-------+
meyer2 | 0.180   | 0.016    | 11.25 |
meyer4 | 1.729   | 0.461    |  3.75 |
-------+---------+----------+-------+