Singular

Hi,
I have the following code to compute the ring of invariants of a derivation:

LIB "ainvar.lib";
ring rw=0,(x(1), y(1), x(2),y(2),z,u),dp;
matrix m[6][1];
m[3,1]=x(1);
m[4,1]=y(1);
m[5,1]=(-x(2)*y(1)+x(1)*y(2)+1)^(1);
m[6,1]=x(2)^2;
ideal in=invariantRing(m,x(2),x(1),0);
in;

At first I tried using the slice x(2)/x(1) in a Pentium processor computer with 8GB of physical memory and with Ubuntu Linux installed but in less than two hours I got a message telling me that there were no more memory (it used the 8GB plus 3GB of swap memory). I tried to do the same in a Windows machine but it gets out of memory after using only 1.5 GB of physical memory.

I followed the Singular algorithm(s) line by line and in the third or fourth computation of further invariants I found that some maps required more memory than the one the computer has to perform some maps.

Next I tried to do the computations using the slice y(2)/y(1) and this time the memory usage is almost always less than 600 MB but it has been processing for one whole week and I still do not receive any feedback (it has been in the third iteration of computation of further invariants the whole week).

I have tried many things such as: perform all computations in the same ring (in 12 variables) from the beginning; follow the algorithm (line by line) doing partial computations, then saving the workspace and deleting all variables not required in that computation; use different computers with different operating systems and different specifications but nothing has worked.

Do you have any suggestion?

Thank you.

Hi,

The size of the Groebner basis computations is just huge, and there are many!
This can be seen with option(prot);

Did you try to compute your examples in positive characteristic, e.g. 32003?
In positive char the GB computations and the maps are rather fast.
Usually the number and the degree of the invariants will be the same as in
char 0 (except for very special primes). This might suffice if you are not
interested in the coefficients of the invariants.

However, if you really need the result in char 0 (and if the computations in
char > 0 are encouraging) you may wish to do modular computations (modStd)
in ainvar.lib. For this you must replace some built-in commands (like preimage)
by procedures which compute these commands directly (as described in
G.-M. Greuel, G. Pfister: A Singular Introduction to Commutative Algebra).

Thank you so much for the information! I actually need the computation in characteristic zero so I will try the modular computations, but I am running the algorithm in prime characteristic to check how many invariants I get and their degrees.
Thank you again!