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Normaliz :: intclMonIdeal(Ideal,allComputations=>...)

intclMonIdeal(Ideal,allComputations=>...) -- normalization of Rees algebra

Synopsis

Description

The leading monomials of the elements of I are considered as generators of a monomial ideal. This function computes the integral closure of I\subset R in the polynomial ring R[t] and the normalization of its Rees algebra. If f_1,\ldots,f_m are the monomial generators of I, then the normalization of the Rees algebra is the integral closure of K[f_1t,\ldots,f_nt] in R[t]. For a definition of the Rees algebra (or Rees ring) see Bruns and Herzog, Cohen-Macaulay rings, Cambridge University Press 1998, p. 182. The function returns the integral closure of the ideal I and the normalization of its Rees algebra. Since the Rees algebra is defined in a polynomial ring with an additional variable, the function creates a new polynomial ring with an additional variable. If the option allComputations is set to true, all data that has been computed by Normaliz is stored in a RationalCone in the CacheTable of the monomial subalgebra returned. This method can also be used with the option grading.

i1 : R=ZZ/37[x,y];
i2 : I=ideal(x^3, x^2*y, y^3, x*y^2);

o2 : Ideal of R
i3 : (intCl,normRees)=intclMonIdeal(allComputations=>true,I)

              3     2   2    3
o3 = (ideal (y , x*y , x y, x ),
                                
                                
                                
                                
                                
     ------------------------------------------------------------------------
     MonomialSubalgebra{cache => CacheTable{...1...}                })
                                           3         2    2      3
                        generators => {y, y a, x, x*y a, x y*a, x a}
                                ZZ
                        ring => --[x..y, a]
                                37

o3 : Sequence
i4 : normRees.cache#"cone"

o4 = RationalCone{cgr => | 0 |                                        }
                         | 4 |
                  equ => | 0 |
                         | 3 |
                  gen => | 0 1 0 |
                         | 0 3 1 |
                         | 1 0 0 |
                         | 1 2 1 |
                         | 2 1 1 |
                         | 3 0 1 |
                  inv => HashTable{ => (1, 1, 1)                     }
                                   class group => 1 : (1)
                                   degree 1 elements => 6
                                   dim max subspace => 0
                                   embedding dim => 3
                                   external index => 1
                                   graded => true
                                   grading denom => 1
                                   grading => (1, 1, -2)
                                   hilbert basis elements => 6
                                   hilbert quasipolynomial denom => 1
                                   hilbert series denom => (1, 1, 1)
                                   hilbert series num => (1, 3)
                                   ideal multiplicity => 9
                                   inhomogeneous => false
                                   integrally closed => true
                                   internal index => 1
                                   multiplicity denom => 1
                                   multiplicity => 4
                                   number extreme rays => 4
                                   number support hyperplanes => 4
                                   primary => true
                                   rank => 3
                                   size triangulation => 4
                                   sum dets => 4
                  sup => | 0 0 1  |
                         | 0 1 0  |
                         | 1 0 0  |
                         | 1 1 -3 |

o4 : RationalCone

Further information

Functions with optional argument named allComputations :