# The generators of the integral closure of the Rees algebra of a monomial ideal.

We use intclMonIdeal to compute the integral closure of a monomial ideal and of its Rees algebra.

 i1 : R=ZZ/37[x_1..x_7]; i2 : I=ideal(x_1..x_6, x_1*x_2*x_3*x_7, x_1*x_2*x_4*x_7, x_1*x_3*x_5*x_7, x_1*x_4*x_6*x_7, x_1*x_5*x_6*x_7, x_2*x_3*x_6*x_7, x_2*x_4*x_5*x_7, x_2*x_5*x_6*x_7,x_3*x_4*x_5*x_7,x_3*x_4*x_6*x_7); o2 : Ideal of R i3 : (intcl,rees)=intclMonIdeal I; i4 : intcl o4 = ideal (x , x , x , x , x , x ) 6 5 4 3 2 1 ZZ o4 : Ideal of --[x ..x , a] 37 1 7 i5 : rees ZZ o5 = --[x , x , x a, x , x a, x , x a, x , x a, x , x a, x , x a] 37 7 6 6 5 5 4 4 3 3 2 2 1 1 ZZ o5 : monomial subalgebra of --[x ..x , a] 37 1 7

The first entry is an ideal, the integral closure of the original ideal, the second one a monomial subalgebra. Each variable in the example appears in a generator of the ideal. Therefore an auxiliary variable a is added to the ring. If there were a free variable in the ring, say x_8, then one can give this variable as a second argument to the function, which then is used as auxiliary variable.

 i6 : R=ZZ/37[x_1..x_8]; i7 : I=ideal(x_1..x_6, x_1*x_2*x_3*x_7, x_1*x_2*x_4*x_7, x_1*x_3*x_5*x_7, x_1*x_4*x_6*x_7, x_1*x_5*x_6*x_7, x_2*x_3*x_6*x_7, x_2*x_4*x_5*x_7, x_2*x_5*x_6*x_7,x_3*x_4*x_5*x_7,x_3*x_4*x_6*x_7); o7 : Ideal of R i8 : (intcl,rees)=intclMonIdeal(I,x_8); i9 : intcl o9 = ideal (x , x , x , x , x , x ) 6 5 4 3 2 1 o9 : Ideal of R i10 : rees ZZ o10 = --[x , x , x x , x , x x , x , x x , x , x x , x , x x , x , x x ] 37 7 6 6 8 5 5 8 4 4 8 3 3 8 2 2 8 1 1 8 o10 : monomial subalgebra of R