[[f1 f2 ...] [t1 t2 ...] [vars params] [k0 k1 limitdeg ]] restriction [ 0-th cohomology group, (-1)-th cohomology group, .... ] [[f1 f2 ...] [t1 t2 ...] [vars params] limitdeg] restriction This function can be used by loading the experimental package cohom.sm1. Restriction of the D-ideal [f1 f2 ...] to t1=0, t2=0, ... is computed. vars is a list of the variables and params is a list of parameters. k0 is the minimum integral root of the b-function and k1 is the maximum integral root of the b-function. If these values are not given and they are small, then they are automatically computed. The program returns 0-th, ..., -limitdeg-th cohomology groups. [vars params] and [k0 k1 deg] are optional arguments. If vars and params are not given, the values of the global variables BFvarlist and BFparlist will be used. For the algorithm, see math.AG/9805006, http://xxx.langl.gov Example 1: cf. math.AG/9801114, Example 1.4 [[(- 2 x Dx - 3 y Dy +1) (3 y Dx^2 - 2 x Dy)] [(x) (y)] [[(x) (y)] [ ]]] restriction :: [ [ 0 , [ ] ] , [ 1 , [ ] ] , [ 1 , [ ] ] ] H^0 = 0, H^(-1)= C^1/(no relation), H^(-2)=C^1/(no relation). Example 2: [[(x Dx-1) (Dy^2)] [(y)] [[(x) (y)] [ ]]] restriction :: [ [ 2 , [ -x*Dx+1 , -x*e*Dx+e ] ] , [ 0 , [ ] ] ] H^0=D_1^2/([-x Dx+1,0],[0, -x Dx + 1]), H^(-1) = 0 where e^0, e^1, e^2, ..., e^(m-1) are standard basis vectors of rank m free module (D_1)^m. D_1 is the ring of differential opertors of one variable x. Example 3: [[(x Dx-1) (Dy^2)] [(y)] [[(x) (y)] [ ]] 0] restriction :: Example 4: [[[(0) (x^2 Dx+x)] [(Dx^2+x Dx^3) (0)]] [(x)] [[(x)] [ ]]] restriction :: In case of vector input, RESTRICTION VARIABLES MUST APPEAR FIRST in the list of variable. We are using wbfRoots to get the roots of b-functions, so we can use only generic weight vector for now.

Nobuki Takayama 2020-11-24