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D.4.19.15 irreddecMon
Procedure from library monomialideal.lib (see monomialideal_lib).
- Usage:
- irreddecMon (I[,alg]); I ideal, alg string.
- Return:
- list, the irreducible components of the monomial ideal I.
(returns -1 if I is not a monomial ideal).
- Assume:
- I is a monomial ideal of the basering k[x(1)..x(n)].
- Note:
- This procedure returns the irreducible decomposition of I.
One may call the procedure with different algorithms using
the optional argument 'alg':
- the direct method following Vasconcelos' book (alg=vas)
- via the Alexander dual and using doble dual (alg=add),
- via the Alexander dual and quotients following E. Miller
(alg=ad),
- the formula of irreducible components (alg=for),
- via the Scarf complex following Milowski (alg=mil),
- using the label algorithm of Roune (alg=lr),
- using the algorithm of Gao-Zhu (alg=gz).
- using the slice algorithm of Roune (alg=sr).
Example:
| | LIB "monomialideal.lib";
ring R = 0,(w,x,y,z),Dp;
ideal I = w^3*x*y,w*x*y*z,x^2*y^2*z^2,x^2*z^4,y^3*z;
// Vasconcelos
irreddecMon (I,"vas");
==> [1]:
==> _[1]=y
==> _[2]=x2
==> [2]:
==> _[1]=w
==> _[2]=z2
==> _[3]=y3
==> [3]:
==> _[1]=y
==> _[2]=z4
==> [4]:
==> _[1]=w
==> _[2]=x2
==> _[3]=y3
==> [5]:
==> _[1]=w
==> _[2]=y2
==> _[3]=z4
==> [6]:
==> _[1]=z
==> _[2]=w3
==> [7]:
==> _[1]=z
==> _[2]=x
==> [8]:
==> _[1]=x
==> _[2]=y3
// Alexander Dual
irreddecMon (I,"ad");
==> [1]:
==> _[1]=w
==> _[2]=y3
==> _[3]=z2
==> [2]:
==> _[1]=w
==> _[2]=y2
==> _[3]=z4
==> [3]:
==> _[1]=x
==> _[2]=z
==> [4]:
==> _[1]=w
==> _[2]=x2
==> _[3]=y3
==> [5]:
==> _[1]=w3
==> _[2]=z
==> [6]:
==> _[1]=x2
==> _[2]=y
==> [7]:
==> _[1]=y
==> _[2]=z4
==> [8]:
==> _[1]=x
==> _[2]=y3
// Scarf Complex
irreddecMon (I,"mil");
==> [1]:
==> _[1]=y
==> _[2]=z4
==> [2]:
==> _[1]=w3
==> _[2]=z
==> [3]:
==> _[1]=w
==> _[2]=y3
==> _[3]=z2
==> [4]:
==> _[1]=w
==> _[2]=y2
==> _[3]=z4
==> [5]:
==> _[1]=w
==> _[2]=x2
==> _[3]=y3
==> [6]:
==> _[1]=x
==> _[2]=y3
==> [7]:
==> _[1]=x2
==> _[2]=y
==> [8]:
==> _[1]=x
==> _[2]=z
// slice algorithm
irreddecMon(I,"sr");
==> [1]:
==> _[1]=y
==> _[2]=z4
==> [2]:
==> _[1]=x2
==> _[2]=y
==> [3]:
==> _[1]=x
==> _[2]=z
==> [4]:
==> _[1]=x
==> _[2]=y3
==> [5]:
==> _[1]=w3
==> _[2]=z
==> [6]:
==> _[1]=w
==> _[2]=y3
==> _[3]=z2
==> [7]:
==> _[1]=w
==> _[2]=y2
==> _[3]=z4
==> [8]:
==> _[1]=w
==> _[2]=x2
==> _[3]=y3
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