Top
Back: normal
Forward: normalC
FastBack: noether_lib
FastForward: normaliz_lib
Up: normal_lib
Top: Singular Manual
Contents: Table of Contents
Index: Index
About: About this document

D.4.15.2 normalP

Procedure from library normal.lib (see normal_lib).

Usage:
normalP(id [,choose]); id = radical ideal, choose = optional list of strings.
Optional parameters in list choose (can be entered in any order):
"withRing", "isPrim", "noFac", "noRed", where
- "noFac" -> factorization is avoided during the computation of the minimal associated primes.
- "isPrim" -> assumes that the ideal is prime. If the assumption does not hold, output might be wrong.
- "withRing" -> the ring structure of the normalization is computed. The number of variables in the new ring is reduced as much as possible.
- "noRed" -> when computing the ring structure, no reduction on the number of variables is done, it creates one new variable for every new module generator of the integral closure in the quotient field.

Assume:
The characteristic of the ground field must be positive. If the option "isPrim" is not set, the minimal associated primes of id are computed first and hence normalP computes the normalization of the radical of id. If option "isPrim" is set, the ideal must be a prime ideal otherwise the result may be wrong.

Return:
a list, say 'nor' of size 2 (resp. 3 if "withRing" is set).
** If option "withRing" is not set:
Only the module structure is computed:
* nor[1] is a list of ideals Ii, i=1..r, in the basering R where r is the number of minimal associated prime ideals P_i of the input ideal id, describing the module structure:
If Ii is given by polynomials g_1,...,g_k in R, then c:=g_k is non-zero in the ring R/P_i and g_1/c,...,g_k/c generate the integral closure of R/P_i as R-module in the quotient field of R/P_i.
* nor[2] shows the delta invariants: it is a list of an intvec of size r, the delta invariants of the r components, and an integer, the total delta invariant of R/id
(-1 means infinite, and 0 that R/P_i resp. R/id is normal).
** If option "withRing" is set:
The ring structure is also computed, and in this case:
* nor[1] is a list of r rings.
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 R/P_i, i.e. isomorphic as K-algebra (K the ground field) to the integral closure of R/P_i in the field of fractions of R/P_i;
- the direct sum of the rings Ri/norid is the normalization of R/id;
- normap gives the normalization map from R to Ri/norid.
* nor[2] gives the module generators of the normalization of R/P_i, it is the same as nor[1] if "withRing" is not set.
* nor[3] shows the delta invariants, it is the same as nor[2] if "withRing" is not set.

Theory:
normalP uses the Leonard-Pellikaan-Singh-Swanson algorithm (using the Frobenius) cf. [A. K. Singh, I. Swanson: An algorithm for computing the integral closure, arXiv:0901.0871].
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. The procedure 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).
Works only in characteristic p > 0; use proc normal in char 0.

Example:
 
See also: normal; normalC.


Top Back: normal Forward: normalC FastBack: noether_lib FastForward: normaliz_lib Up: normal_lib Top: Singular Manual Contents: Table of Contents Index: Index About: About this document
            User manual for Singular version 3-1-6, Dec 2012, generated by texi2html.