The leading monomials of the elements of I are considered as generators of a monomial ideal. This function computes the integral closure of I⊂R in the polynomial ring R[t] and the normalization of its Rees algebra. If f
1,...,f
m are the monomial generators of I, then the normalization of the Rees algebra is the integral closure of K[f
1t,...,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 3 2
o3 = (ideal (x , x y, y , x*y ),
------------------------------------------------------------------------
MonomialSubalgebra{cache => CacheTable{...1...} })
3 2 3 2
generators => {x, y, x a, x y*a, y a, x*y a}
ZZ
ring => --[x, y, a]
37
o3 : Sequence
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i4 : normRees.cache#"cone"
o4 = RationalCone{cgr => 0 }
equ => 0
gen => | 1 0 0 |
| 0 1 0 |
| 3 0 1 |
| 2 1 1 |
| 0 3 1 |
| 1 2 1 |
inv => HashTable{ => (1, 1, 1) }
degree 1 elements => 6
embedding dim => 3
graded => true
grading => (1, 1, -2)
grading denom => 1
hilbert basis elements => 6
hilbert quasipolynomial denom => 1
hilbert series denom => (1, 1, 1)
hilbert series num => (1, 3)
ideal multiplicity => 9
index => 1
inhomogeneous => false
multiplicity => 4
multiplicity denom => 1
number extreme rays => 4
number support hyperplanes => 4
primary => true
rank => 3
size triangulation => 3
sum dets => 4
sup => | 0 1 0 |
| 0 0 1 |
| 1 1 -3 |
| 1 0 0 |
o4 : RationalCone
|