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# reesMap -- compute the Rees map of $I$

## Synopsis

• Usage:
reesMap I
• Inputs:
• I, an ideal, a homogeneous ideal of polynomial ring $S = \mathbb{K}[x_1,\ldots,x_n]$, $\mathbb{K}$ a field
• Outputs:
• g, , the Rees map of $I$

## Description

Given a homogeneous ideal $I$, with minimal homogeneous generating set $\{f_1,\dots,f_m\}$, this method yields the bigraded map $g: \mathbb{K}[x_1,\dots,x_n,y_1,\dots,y_m] \to \mathbb{K}[x_1,\dots,x_n,t]$ defined by setting $g(x_i) = x_i$ for $1 \le i \le n$ and $g(y_j) = f_j t$ for $1 \le j \le m$ and with $\text{bideg}(x_i) = (1,0)$ and $\text{bideg}(y_j) = (\text{deg}(f_j), 1)$.

 i1 : S = QQ[x_1..x_3]; i2 : I = ideal(x_1*x_2,x_1*x_3,x_2*x_3); o2 : Ideal of S i3 : reesMap I o3 = map (QQ[x ..x , t], QQ[x ..x , y ..y ], {x , x , x , x x t, x x t, x x t}) 1 3 1 3 1 3 1 2 3 1 2 1 3 2 3 o3 : RingMap QQ[x ..x , t] <-- QQ[x ..x , y ..y ] 1 3 1 3 1 3
 i4 : S = QQ[x_1..x_4]; i5 : I = ideal(x_1^3*x_2+x_3^4,x_1+x_2+x_4,x_3^3); o5 : Ideal of S i6 : reesMap I 3 4 3 o6 = map (QQ[x ..x , t], QQ[x ..x , y ..y ], {x , x , x , x , x x t + x t, x t + x t + x t, x t}) 1 4 1 4 1 3 1 2 3 4 1 2 3 1 2 4 3 o6 : RingMap QQ[x ..x , t] <-- QQ[x ..x , y ..y ] 1 4 1 4 1 3

## Ways to use reesMap :

• reesMap(Ideal)

## For the programmer

The object reesMap is .