##### 7.5.6.0. BernsteinSatoIdeal

Procedure from library `dmodideal.lib` (see dmodideal_lib).

Usage:
  BernsteinSatoIdeal(F [,eng,met,us]); F an ideal, eng, us optional ints, met optional int or intvec

Return:
  ring

Purpose:
  compute two kinds of Bernstein-Sato ideals, associated to f = F[1]*..*F[P], with the multivariate algorithm by Briancon and Maisonobe.

Assume:
  basering is a commutative polynomial ring in characteristic 0

Note:
  activate the output ring with the `setring` command. In this ring,
  - the ideal LD is the annihilator of F[1]^s_1*..*F[P]^s_p,
  - the list or ideal BS is a Bernstein-Sato ideal of a polynomial f = F[1]*..*F[P]. If eng <>0, `std` is used for Groebner basis computations, otherwise, and by default `slimgb` is used. If met <0, the B-Sigma ideal (cf. Castro and Ucha, 'On the computation of Bernstein-Sato ideals', 2005) is computed. If 0 < met < P, then the ideal B_P (cf. the paper) is computed. If met is an intvec, Budurs generalized Bernstein-Sato ideal associated to met is computed.
  Otherwise, and by default, the ideal B (cf. the paper) is computed. If us<>0, then syzygies-driven method is used.
  If printlevel=1, progress debug messages will be printed, if printlevel>=2, all the debug messages will be printed.

Example:

    LIB "dmodideal.lib";
    ring R = 0,(x,y),dp;
    ideal F = x^2-y,y;
    // first we compute the ideal B:
    def S = BernsteinSatoIdeal(F);
    setring S;
    BS;
    ==> BS[1]=4*s(1)^3*s(2)+4*s(1)^3+8*s(1)^2*s(2)^2+28*s(1)^2*s(2)+20*s(1)^2+4*s\
       (1)*s(2)^3+28*s(1)*s(2)^2+55*s(1)*s(2)+31*s(1)+4*s(2)^3+20*s(2)^2+31*s(2)\
       +15
    // secondly we compute the ideal B_1:
    setring R;
    def S = BernsteinSatoIdeal(F,0,1);
    ==> // ** redefining S (def S = BernsteinSatoIdeal(F,0,1);) ./examples/Bernst\
       einSatoIdeal.sing:10
    setring S;
    BS;
    ==> BS[1]=2*s(1)^2+2*s(1)*s(2)+5*s(1)+2*s(2)+3
    // thirdly we compute the ideal B_sigma:
    setring R;
    def S = BernsteinSatoIdeal(F,0,-1);
    ==> // ** redefining S (def S = BernsteinSatoIdeal(F,0,-1);) ./examples/Berns\
       teinSatoIdeal.sing:15
    setring S;
    BS;
    ==> BS[1]=2*s(1)*s(2)+2*s(1)+2*s(2)^2+5*s(2)+3
    ==> BS[2]=2*s(1)^2+3*s(1)-2*s(2)^2-3*s(2)

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