# Singular

##### 7.5.7.0. bfctVarIn
Procedure from library `dmodvar.lib` (see dmodvar_lib).

Usage:
bfctVarIn(I [,a,b,c]); I an ideal, a,b,c optional ints

Return:
list of ideal and intvec

Purpose:
computes the roots of the Bernstein-Sato polynomial and their multiplicities for an affine algebraic variety defined by I.

Assume:
The basering is commutative and over a field of characteristic 0.
Varnames of the basering do not include t(1),...,t(r) and Dt(1),...,Dt(r), where r is the number of entries of the input ideal.

Note:
In the output list, say L,
- L[1] of type ideal contains all the rational roots of a b-function,
- L[2] of type intvec contains the multiplicities of above roots,
- optional L[3] of type string is the part of b-function without rational roots.
Note, that a b-function of degree 0 is encoded via L[1][1]=0, L[2]=0 and L[3] is 1 (for nonzero constant) or 0 (for zero b-function).
If a<>0, the ideal is used as given. Otherwise, and by default, a heuristically better suited generating set is used to reduce computation time.
If b<>0, `std` is used for GB computations in characteristic 0, otherwise, and by default, `slimgb` is used.
If c<>0, a matrix ordering is used for GB computations, otherwise, and by default, a block ordering is used.
Further note, that in this proc, the initial ideal of the multivariate Malgrange ideal defined by I is computed and then a system of linear equations is solved by linear reductions following the ideas by Noro.
The result is shifted by 1-codim(Var(F)) following (BMS06).

Display:
If printlevel=1, progress debug messages will be printed,
if printlevel>=2, all the debug messages will be printed.

Example:
 ```LIB "dmodvar.lib"; ring R = 0,(x,y,z),dp; ideal F = x^2+y^3, z; list L = bfctVarIn(F); L; ==> [1]: ==> _[1]=-5/6 ==> _[2]=-1 ==> _[3]=-7/6 ==> [2]: ==> 1,1,1 ```