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

## Synopsis

• Usage:
intclMonIdeal I
• Inputs:
• an ideal, the leading monomials of the elements of the ideal are considered as generators of a monomial ideal
• Outputs:
• an ideal, the integral closure of the input ideal
• an instance of the type MonomialSubalgebra, the normalization of the Rees algebra of I

## 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