2.viii. Complexity of these computations
A measure of the complexity of symbolic computation is the degree of the
system (number of solutions) and the dimension of the ambient variety
(number of variables).
Also useful are the size and time it takes to compute a (particular)
Gröbner basis.
The table below displays, for each small m,p the degree,
dm,p of the system, the time it takes to compute a degree
reverse lexicographic Gröbner basis, the size of that basis, and number
of generators.
These were all computed using
Singular-1.2.1 on
a K6/2 300MHz processor with 256M memory running Linux.
The data m,p are each linked to a Maple V.5 script that generates a
Singular script that calculates a degree reverse lexicographic Gröbner
basis.
The Singular script is viewed by clicking on the time entry.
These computations were done in local coordinates for the Grassmannian;
while there are 2 more variables than the used for the verifications
in Section 2.vii,
the degree reverse lexicographic Gröbner basis computation is
considerably faster with this choice of local coordinates.
Computing Degree Reverse Lexicographic
Gröbner Bases
| m,p
| 4,2
| 2,4
| 5,2
| 2,5
| 3,3
| 6,2
| 2,6
| 7,2
|
|---|
| dm,p
| 14 | 14
| 42 | 42
| 42 | 132
| 132 | 429
|
|---|
| time(sec)
| .04
| .02
| 1.45
| 3.69
| 1.50
| 78.6
| 509.8
| 8175
|
|---|
| Size
| 1.3K | 1.1K
| 12.8K | 15.4K
| 18.6K | 202K
| 368K | 4.58M
|
|---|
| # Generators
| 20 | 18
| 47 | 43
| 43 | 120
| 127 | 334
|
|---|