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



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]

o5 : monomial subalgebra of R

Ways to use this method: