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D.4.23.3 normalC
Procedure from library normal.lib (see normal_lib).
- Usage:
- normalC(id [,choose]); id = radical ideal, choose = optional list
of string.
Optional parameters in list choose (can be entered in any order):
Decomposition:
- "equidim" -> computes first an equidimensional decomposition,
and then the normalization of each component (default).
- "prim" -> computes first the minimal associated primes, and then
the normalization of each prime.
- "noDeco" -> no preliminary decomposition is done. If the ideal is
not equidimensional radical, output might be wrong.
- "isPrim" -> assumes that the ideal is prime. If the assumption does
not hold, output might be wrong.
- "noFac" -> factorization is avoided in the computation of the
minimal associated primes;
Other:
- "withGens" -> the minimal associated primes P_i of id are
computed and for each P_i, algebra generators of the integral closure
of basering/P_i are computed as elements of its quotient field;
If choose is not given or empty, the default options are used.
- Assume:
- The ideal must be radical, for non-radical ideals the output may
be wrong (id=radical(id); makes id radical). However, if option
"prim" is set the minimal associated primes are computed first
and hence normalC computes the normalization of the radical of id.
"isPrim" should only be used if id is known to be irreducible.
- Return:
- a list, say nor, of size 2 (resp. 3 if option "withGens" is set).
* nor[1] is always a of r rings, where r is the number of associated
primes with option "prim" (resp. >= no of equidimenensional
components with option "equidim").
Each ring Ri=nor[1][i], i=1..r, contains two ideals with given
names norid and normap such that
- Ri/norid is the normalization of the i-th component, i.e. the
integral closure in its field of fractions as affine ring, i.e. Ri is
given in the form K[X(1..p),T(1..q)], where K is the ground field;
- normap gives the normalization map from basering/id to
Ri/norid for each i (the j-th element of normap is mapped to the
j-th variable of R).
- the direct sum of the rings Ri/norid is the normalization
of basering/id;
** If option "withGens" is not set:
* nor[2] shows the delta invariants: nor[2] is a list of an intvec of
size r, the delta invariants of the r components, and an integer, the
delta invariant of basering/id. (-1 means infinite, 0 that basering/P_i
resp. basering/input is normal, -2 means that delta resp. delta of one
of the components is not computed (which may happen if "equidim" is
given).
** If option "withGens" is set:
* nor[2] is a list of ideals Ii=nor[2][i], i=1..r, in the basering,
generating the integral closure of basering/P_i in its quotient field
as K-algebra (K the ground field):
If Ii is given by polynomials g_1,...,g_k, then c:=g_k is a non-zero
divisor and the j-th variables of the ring Ri satisfies var(j)=g_j/c,
j=1..k-1, as element in the quotient field of basering/P_i. The
g_j/g_k+1 are K-algebra generators of the integral closure of
basering/P_i.
* nor[3] shows the delta invariant as above.
- Theory:
- We use the Grauert-Remmert-de Jong algorithm [c.f. G.-M. Greuel,
G. Pfister: A SINGULAR Introduction to Commutative Algebra, 2nd Edition.
Springer Verlag (2007)].
The procedure computes the algebra structure and the delta invariant of
the normalization of R/id:
The normalization is an affine algebra over the ground field K
and nor[1] presents it as such: Ri = K[X(1..p),T(1..q)] and Ri/norid
is the integral closure of R/P_i; if option "withGens" is set the
X(j) and T(j) are expressed as quotients in the total ring of
fractions. Note that the X(j) and T(j) generate the integral closure
as K-algebra, but not necessarily as R-module (since relations of the
form X(1)=T(1)*T(2) may have been eliminated). Geometrically the
algebra structure is relevant since the variety of the ideal norid in
Ri is the normalization of the variety of the ideal P_i in R.
The delta invariant of a reduced ring A is dim_K(normalization(A)/A).
For A=K[x1,...,xn]/id we call this number also the delta invariant of
id. nor[3] returns the delta invariants of the components P_i and of
id.
- Note:
- To use the i-th ring type:
def R=nor[1][i]; setring R;.
Increasing/decreasing printlevel displays more/less comments
(default: printlevel=0).
Not implemented for local or mixed orderings or quotient rings.
For local or mixed orderings use proc 'normal'.
If the input ideal id is weighted homogeneous a weighted ordering may
be used (qhweight(id); computes weights).
Example:
| | LIB "normal.lib";
printlevel = printlevel+1;
ring s = 0,(x,y),dp;
ideal i = (x2-y3)*(x2+y2)*x;
list nor = normalC(i);
==> // We use method 'prim'
==>
==> // number of irreducible components: 3
==>
==> // computing the normalization of component 1
==> ----------------------------------------
==> // delta of component 1
==> 1
==>
==> // computing the normalization of component 2
==> ----------------------------------------
==> // delta of component 2
==> 1
==>
==> // computing the normalization of component 3
==> ----------------------------------------
==> // delta of component 3
==> 0
==> // Sum of delta for all components
==> 2
==> // Compute intersection multiplicities of the components
==>
==> // 'normalC' created a list, say nor, of two lists:
==> // To see the result, type
==> nor;
==>
==> // * nor[1] is a list of 3 ring(s).
==> // To access the i-th ring nor[1][i] give it a name, say Ri, and type e.g\
.
==> def R1 = nor[1][1]; setring R1; norid; normap;
==> // and similair for the other rings nor[1][i];
==> // Ri/norid is the affine algebra of the normalization of r/P_i (where P\
_i
==> // is an associated prime or an equidimensional part of the input ideal i\
d)
==> // and normap the normalization map from the basering to Ri/norid;
==>
==> // * nor[2] shows the delta-invariant of each component and of id
==> // (-1 means infinite, 0 that r/P_i resp. r/id is normal, and -2 that del\
ta
==> // of a component was not computed).
nor;
==> [1]:
==> [1]:
==> // coefficients: QQ
==> // number of vars : 1
==> // block 1 : ordering dp
==> // : names T(1)
==> // block 2 : ordering C
==> [2]:
==> // coefficients: QQ
==> // number of vars : 2
==> // block 1 : ordering dp
==> // : names T(1) T(2)
==> // block 2 : ordering C
==> [3]:
==> // coefficients: QQ
==> // number of vars : 1
==> // block 1 : ordering dp
==> // : names T(1)
==> // block 2 : ordering C
==> [2]:
==> [1]:
==> 0,1,1
==> [2]:
==> 13
// 2 branches have delta = 1, and 1 branch has delta = 0
// the total delta invariant is 13
def R2 = nor[1][2]; setring R2;
norid; normap;
==> norid[1]=T(2)^2+1
==> normap[1]=-T(1)*T(2)
==> normap[2]=T(1)
printlevel = printlevel-1;
ring r = 2,(x,y,z),dp;
ideal i = z3-xy4;
nor = normalC(i); nor;
==>
==> // 'normalC' created a list, say nor, of two lists:
==> // To see the result, type
==> nor;
==>
==> // * nor[1] is a list of 1 ring(s).
==> // To access the i-th ring nor[1][i] give it a name, say Ri, and type e.g\
.
==> def R1 = nor[1][1]; setring R1; norid; normap;
==> // and similair for the other rings nor[1][i];
==> // Ri/norid is the affine algebra of the normalization of r/P_i (where P\
_i
==> // is an associated prime or an equidimensional part of the input ideal i\
d)
==> // and normap the normalization map from the basering to Ri/norid;
==>
==> // * nor[2] shows the delta-invariant of each component and of id
==> // (-1 means infinite, 0 that r/P_i resp. r/id is normal, and -2 that del\
ta
==> // of a component was not computed).
==> [1]:
==> [1]:
==> // coefficients: ZZ/2
==> // number of vars : 3
==> // block 1 : ordering dp
==> // : names T(1) T(2) T(3)
==> // block 2 : ordering C
==> [2]:
==> [1]:
==> -1
==> [2]:
==> -1
// the delta invariant is infinite
// xy2z/z2 and xy3/z2 generate the integral closure of r/i as r/i-module
// in its quotient field Quot(r/i)
// the normalization as affine algebra over the ground field:
def R = nor[1][1]; setring R;
norid; normap;
==> norid[1]=T(3)^3+T(1)*T(2)
==> normap[1]=T(1)
==> normap[2]=T(2)
==> normap[3]=T(2)*T(3)
setring r;
nor = normalC(i, "withGens", "prim"); // a different algorithm
==>
==> // 'normalC' created a list, say nor, of three lists:
==> // To see the list type
==> nor;
==>
==> // * nor[1] is a list of 1 ring(s)
==> // To access the i-th ring nor[1][i] give it a name, say Ri, and type e.g\
.
==> def R1 = nor[1][1]; setring R1; norid; normap;
==> // For the other rings type first (if R is the name of your original base\
ring)
==> setring R;
==> // and then continue as for R1.
==> // Ri/norid is the affine algebra of the normalization of the i-th
==> // component R/P_i (where P_i is an associated prime of the input ideal i\
d)
==> // and normap the normalization map from R to Ri/norid.
==>
==> // * nor[2] is a list of 1 ideal(s), each ideal nor[2][i] consists of
==> // elements g1..gk of R such that the gj/gk generate the integral
==> // closure of R/P_i as sub-algebra in the quotient field of R/P_i, with
==> // gj/gk being mapped by normap to the j-th variable of Ri;
==>
==> // * nor[3] shows the delta-invariant of each component and of id
==> // (-1 means infinite, and 0 that R/P_i resp. R/id is normal).
nor;
==> [1]:
==> [1]:
==> // coefficients: ZZ/2
==> // number of vars : 6
==> // block 1 : ordering dp
==> // : names X(1) X(2) X(3) X(4) T(2) T(3)
==> // block 2 : ordering C
==> [2]:
==> [1]:
==> _[1]=x2y3
==> _[2]=z3
==> _[3]=xy3z
==> _[4]=xy2z2
==> _[5]=xyz2
==> _[6]=xy2z
==> _[7]=xy3
==> [3]:
==> [1]:
==> -1
==> [2]:
==> -1
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See also:
normal;
normalP.
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