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# intclMonIdeal(Ideal,RingElement) -- normalization of Rees algebra

## Synopsis

• Function: intclMonIdeal
• Usage:
intclMonIdeal(I,t)
• Inputs:
• an ideal, the leading monomials of the elements of the ideal are considered as generators of a monomial ideal
• the free variable of the ring that should be used for the computation (optional),
• Optional inputs:
• allComputations => ..., default value false, normalization of Rees algebra
• grading => ..., default value {}, normalization of Rees algebra
• 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. If there is a free variable in the original ring (i.e. a variable that does not appear in any of the generators of I), you can give the function that variable as second input. The function then uses it instead of creating a new one. Note that in this case the input ideal is considered as ideal in the smaller polynomial ring.
 i1 : R=ZZ/37[x,y,t]; i2 : I=ideal(x^3, x^2*y, y^3, x*y^2); o2 : Ideal of R i3 : (intCl,normRees)=intclMonIdeal(I,t); i4 : intCl 3 2 2 3 o4 = ideal (y , x*y , x y, x ) o4 : Ideal of R i5 : normRees ZZ 3 2 2 3 o5 = --[y, y t, x, x*y t, x y*t, x t] 37 o5 : monomial subalgebra of R