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];
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i2 : I=ideal(x^3, x^2*y, y^3, x*y^2);
o2 : Ideal of R
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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
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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
|