Singular

D.4.1.2 absFactorizeBCG

Usage:

absFactorizeBCG(p [,s]); p poly, s string

Assume:

coefficient field is the field of rational numbers or a transcendental extension thereof

Return:

ring R which is obtained from the current basering by adding a new parameter (if a string s is given as a second input, the new parameter gets this string as name). The ring R comes with a list absolute_factors with the following entries:

absolute_factors[1]: ideal (the absolute factors) absolute_factors[2]: intvec (the multiplicities) absolute_factors[3]: ideal (the minimal polynomials) absolute_factors[4]: int (total number of nontriv. absolute factors)

The entry absolute_factors[1][1] is a constant, the entry absolute_factors[3][1] is the parameter added to the current ring.
Each of the remaining entries absolute_factors[1][j] stands for a class of conjugated absolute factors. The corresponding entry absolute_factors[3][j] is the minimal polynomial of the field extension over which the factor is minimally defined (its degree is the number of conjugates in the class). If the entry absolute_factors[3][j] coincides with absolute_factors[3][1], no field extension was necessary for the jth (class of) absolute factor(s).

Note:

All factors are presented denominator- and content-free. The constant factor (first entry) is chosen such that the product of all (!) the (denominator- and content-free) absolute factors of p equals p / absolute_factors[1][1].

LIB "absfact.lib";
ring R = (0), (x,y), lp;
poly p = (-7*x^2 + 2*x*y^2 + 6*x + y^4 + 14*y^2 + 47)*(5x2+y2)^3*(x-y)^4;
def S = absFactorizeBCG(p) ;
==> 
==> // 'absFactorizeBCG' created a ring, in which a list absolute_factors (th\
   e
==> // absolute factors) is stored.
==> // To access the list of absolute factors, type (if the name S was assign\
   ed
==> // to the return value):
==> //        setring(S); absolute_factors;
==>   
setring(S);
absolute_factors;
==> [1]:
==>    _[1]=1
==>    _[2]=(2a-13)*x+y2+(a)
==>    _[3]=-x+y
==>    _[4]=(-a)*x+y
==> [2]:
==>    1,1,4,3
==> [3]:
==>    _[1]=(a)
==>    _[2]=(a2-14a+47)
==>    _[3]=(a)
==>    _[4]=(a2+5)
==> [4]:
==>    12