Singular

D.4.27.6 syModStd

Usage:

syModStd(I,sigma); I ideal, sigma intvec

Assume:

size(sigma) = nvars(basering) and sigma(I) = I. If size(#) > 0, then # contains either 1, 2 or 4 integers such that
- #[1] is the number of available processors for the computation,
- #[2] is an optional parameter for the exactness of the computation, if #[2] = 1, the procedure computes a standard basis for sure,
- #[3] is the number of primes until the first lifting,
- #[4] is the constant number of primes between two liftings until the computation stops.

Return:

ideal, a standard basis of I if no warning appears;

Note:

The procedure computes a standard basis of the ideal I (over the rational numbers) by using modular methods and the fact that I is invariant under the variable permutation sigma.
By default the procedure computes a standard basis of I for sure, but if the optional parameter #[2] = 0, it computes a standard basis of I with high probability.
The procedure distinguishes between different variants for the standard basis computation in positive characteristic depending on the ordering of the basering, the parameter #[2] and if the ideal I is homogeneous.
- variant = 1, if I is homogeneous,
- variant = 2, if I is not homogeneous, 1-block-ordering,
- variant = 3, if I is not homogeneous, complicated ordering (lp or > 1 block),
- variant = 4, if I is not homogeneous, ordering lp, dim(I) = 0.

LIB "symodstd.lib";
ring R1 = 0, (x,y,z), dp;
ideal I;
I[1] = -2xyz4+xz5+xz;
I[2] = -2xyz4+yz5+yz;
intvec sigma = 2,1,3;
ideal sI = syModStd(I,sigma);
sI;
==> sI[1]=x2yz-xy2z
==> sI[2]=xz5-yz5+xz-yz
==> sI[3]=xyz4-1/2yz5-1/2yz
==> sI[4]=y2z5-1/2yz6-xyz+y2z-1/2yz2
ring R2 = 0, x(1..4), dp;
ideal I = cyclic(4);
I;
==> I[1]=x(1)+x(2)+x(3)+x(4)
==> I[2]=x(1)*x(2)+x(2)*x(3)+x(1)*x(4)+x(3)*x(4)
==> I[3]=x(1)*x(2)*x(3)+x(1)*x(2)*x(4)+x(1)*x(3)*x(4)+x(2)*x(3)*x(4)
==> I[4]=x(1)*x(2)*x(3)*x(4)-1
intvec pi = 2,3,4,1;
ideal sJ1 = syModStd(I,pi,1);
ideal sJ2 = syModStd(I,pi,1,0);
size(reduce(sJ1,sJ2));
==> 0
size(reduce(sJ2,sJ1));
==> 0