the capacity of Groebner Basis functions

I am just interested in the performance or efficiency of Greobner Basis functions.

If the ideal has thousands of variables and thousands of polynomials, whether the Groebner Basis computing functions can handle such a large scale problem.

Another question:
If I have thousands of variables and thousands of polynomials in an ideal, how I can put them to the Groebner Basis computing functions.


Thanks a lot.

In general, Gröbner bases computations are of double-exponential worst-case complexity. So, this would make it impossible to apply them to your problems.
But, for a large amount of problem classes there exist more efficient algorithms. SINGULARs command uses a sophisticated heuristic of selecting appropriate subroutines. It might help, if one has some background information about the problem. In that case one can find an optimized routine, or even a good preprocessing.

About, calling Singular. The maximum number of variables in Singular is 32767. First you define a ring like this:

You may also name each variable:

For the polynomial system in question you write something like


Of course, for thousands of variables you should write everything in a file and call

from the command line.

Regards,
Alexander

I would also recommend first to scale down your problem, i.e. reduce the number of equations and variables in the way that general behavior is supposedly preserved. You can run then GB-computations and see what comes out. This may hint you on some special methods of handling your bigger system.

Thank you, but I still want to handle the problem by reading it from file and send it to singular.
Could you tell me what the files related to compute Groebner Basis are? So maybe I can make some modifications to meet the requirements of my problem.

Much appreciation.