# joinIdeals -- Computes the join of two ideals

## Synopsis

• Usage:
K = joinIdeals(I, J)
• Inputs:
• Outputs:
• K, an ideal, the join of the ideals I and J

## Description

This method computes the join of two ideals. The join can be used to describe very interesting types of primary ideals that include the symbolic powers of prime ideals. For more details reader is referred to Section 7 of the paper Primary ideals and their differential equations.

 i1 : R = QQ[x_1..x_9] o1 = R o1 : PolynomialRing i2 : MM = genericMatrix(R, 3, 3) o2 = | x_1 x_4 x_7 | | x_2 x_5 x_8 | | x_3 x_6 x_9 | 3 3 o2 : Matrix R <--- R i3 : P = minors(2, MM) o3 = ideal (- x x + x x , - x x + x x , - x x + x x , - x x + x x , - 2 4 1 5 3 4 1 6 3 5 2 6 2 7 1 8 ------------------------------------------------------------------------ x x + x x , - x x + x x , - x x + x x , - x x + x x , - x x + x x ) 3 7 1 9 3 8 2 9 5 7 4 8 6 7 4 9 6 8 5 9 o3 : Ideal of R i4 : M = ideal(x_1^2, x_2^2, x_3^2, x_4, x_5, x_6, x_7, x_8, x_9) 2 2 2 o4 = ideal (x , x , x , x , x , x , x , x , x ) 1 2 3 4 5 6 7 8 9 o4 : Ideal of R i5 : Q = joinIdeals(P, M) 3 3 2 2 o5 = ideal (x x - x x , x x - x x , x x - x x , x x - 3x x x x + 6 8 5 9 6 7 4 9 5 7 4 8 3 8 2 3 8 9 ------------------------------------------------------------------------ 2 2 3 3 3 2 2 2 2 3 2 2 2 3x x x x - x x , x x x - 3x x x x x + 3x x x x - x x x , x x x x - 2 3 8 9 2 9 3 5 8 2 3 5 8 9 2 3 5 9 2 6 9 2 3 7 8 ------------------------------------------------------------------------ 2 2 2 2 2 2 2 2 2 2 2 x x x x - x x x x + x x x x + x x x x - x x x x , x x x x x - 1 3 7 8 2 3 7 9 1 3 8 9 1 2 7 9 1 2 8 9 2 3 4 7 8 ------------------------------------------------------------------------ 2 2 2 2 2 2 2 2 3 2 x x x x - x x x x x + x x x x x + x x x x - x x x x , x x x - 1 3 4 8 2 3 4 7 9 1 3 5 8 9 1 2 4 9 1 2 5 9 3 5 8 ------------------------------------------------------------------------ 2 2 2 3 2 2 2 2 2 2 3x x x x + 3x x x x x - x x x , x x x x - x x x x x - x x x x + 2 3 5 9 2 3 5 6 9 2 6 9 2 3 4 8 1 3 4 5 8 2 3 4 9 ------------------------------------------------------------------------ 2 2 2 2 3 3 2 2 2 2 3 3 x x x x + x x x x x - x x x x x , x x - 3x x x x + 3x x x x - x x , 1 3 5 9 1 2 4 6 9 1 2 5 6 9 3 7 1 3 7 9 1 3 7 9 1 9 ------------------------------------------------------------------------ 3 3 2 2 2 2 3 3 3 2 2 2 2 x x - 3x x x x + 3x x x x - x x , x x x - 3x x x x x + 3x x x x - 2 7 1 2 7 8 1 2 7 8 1 8 3 4 7 1 3 4 7 9 1 3 4 9 ------------------------------------------------------------------------ 3 2 3 2 2 2 2 3 2 3 2 2 2 x x x , x x x - 3x x x x x + 3x x x x - x x x , x x x - 3x x x x + 1 6 9 2 4 7 1 2 4 7 8 1 2 4 8 1 5 8 3 4 7 1 3 4 9 ------------------------------------------------------------------------ 2 3 2 3 2 2 2 2 3 2 3 3 3x x x x x - x x x , x x x - 3x x x x + 3x x x x x - x x x , x x - 1 3 4 6 9 1 6 9 2 4 7 1 2 4 8 1 2 4 5 8 1 5 8 3 5 ------------------------------------------------------------------------ 2 2 2 2 3 3 2 2 2 2 2 2 2 2 3x x x x + 3x x x x - x x , x x x x - x x x x - x x x x + x x x x 2 3 5 6 2 3 5 6 2 6 2 3 4 5 1 3 4 5 2 3 4 6 1 3 5 6 ------------------------------------------------------------------------ 2 2 2 2 3 3 2 2 2 2 3 3 3 3 + x x x x - x x x x , x x - 3x x x x + 3x x x x - x x , x x - 1 2 4 6 1 2 5 6 3 4 1 3 4 6 1 3 4 6 1 6 2 4 ------------------------------------------------------------------------ 2 2 2 2 3 3 2 3 2 2 4 3 2 3x x x x + 3x x x x - x x , x x x x x - x x x + x x x x x - 1 2 4 5 1 2 4 5 1 5 1 2 3 7 8 1 3 8 2 3 7 8 9 ------------------------------------------------------------------------ 2 2 2 3 4 2 2 3 2 2 2 2 2 4x x x x x x + 3x x x x x - x x x + 3x x x x x - 2x x x x , 1 2 3 7 8 9 1 2 3 8 9 2 7 9 1 2 7 8 9 1 2 8 9 ------------------------------------------------------------------------ 2 3 2 2 3 3 2 2 2 2 x x x x x - x x x x + x x x x x x - 4x x x x x x + 3x x x x x x - 1 2 3 4 8 1 3 5 8 2 3 4 7 8 9 1 2 3 4 8 9 1 2 3 5 8 9 ------------------------------------------------------------------------ 4 2 3 2 2 2 2 4 2 2 3 2 2 2 2 2 x x x x + 3x x x x x - 2x x x x x , x x x - 4x x x x x - x x x x + 2 4 7 9 1 2 4 8 9 1 2 5 8 9 3 7 8 1 3 7 8 9 2 3 7 9 ------------------------------------------------------------------------ 2 2 2 2 2 2 2 3 2 2 4 3 2 2 2 2 3 4x x x x x x + 3x x x x - 4x x x x x + x x x , x x x x + x x x x - 1 2 3 7 8 9 1 3 8 9 1 2 3 8 9 1 2 9 1 3 7 8 2 3 7 9 ------------------------------------------------------------------------ 2 2 2 2 2 2 2 2 3 2 2 2 2 3 4x x x x x - 4x x x x x + 4x x x x x x + 3x x x x + 3x x x x - 1 3 7 8 9 1 2 3 7 9 1 2 3 7 8 9 1 3 8 9 1 2 7 9 ------------------------------------------------------------------------ 3 3 4 2 3 2 2 2 2 2 2 4x x x x , x x x x - 4x x x x x - x x x x x + 4x x x x x x + 1 2 8 9 3 4 7 8 1 3 4 8 9 2 3 4 7 9 1 2 3 4 8 9 ------------------------------------------------------------------------ 2 2 2 2 3 2 2 3 3 2 2 2 2 3x x x x x - 4x x x x x + x x x x , x x x x x + x x x x x - 1 3 5 8 9 1 2 3 5 9 1 2 6 9 1 3 4 7 8 2 3 4 7 9 ------------------------------------------------------------------------ 2 2 2 2 2 2 2 3 2 2 2 3 4x x x x x - 4x x x x x x + 4x x x x x x + 3x x x x x + 3x x x x - 1 3 4 8 9 1 2 3 4 7 9 1 2 3 4 8 9 1 3 5 8 9 1 2 4 9 ------------------------------------------------------------------------ 3 3 2 2 2 2 2 2 3 2 2 4x x x x , x x x x x x - x x x x + x x x x x - 4x x x x x x x + 1 2 5 9 1 2 3 4 5 8 1 3 5 8 2 3 4 8 9 1 2 3 4 5 8 9 ------------------------------------------------------------------------ 2 2 4 2 2 3 2 2 2 2 2 4 2 2 3 3x x x x x x - x x x + 3x x x x x - 2x x x x , x x x - 4x x x x x x 1 2 3 5 8 9 2 4 9 1 2 4 5 9 1 2 5 9 3 4 8 1 3 4 5 8 9 ------------------------------------------------------------------------ 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 - x x x x + 4x x x x x x + 3x x x x - 4x x x x x x + x x x x , 2 3 4 9 1 2 3 4 5 9 1 3 5 9 1 2 3 5 6 9 1 2 6 9 ------------------------------------------------------------------------ 3 2 2 2 2 2 2 2 2 2 2 2 2 x x x x + x x x x x - 4x x x x x x - 4x x x x x + 4x x x x x x + 1 3 4 8 2 3 4 7 9 1 3 4 5 8 9 1 2 3 4 9 1 2 3 4 5 9 ------------------------------------------------------------------------ 3 2 2 2 2 2 3 2 2 2 2 2 3 3x x x x + 3x x x x x - 4x x x x x , x x x x x x - x x x x + 1 3 5 9 1 2 4 6 9 1 2 5 6 9 1 2 3 4 5 8 1 3 5 8 ------------------------------------------------------------------------ 3 2 2 2 2 3 4 2 3 x x x x x - 4x x x x x x + 3x x x x x - x x x x + 3x x x x x x - 2 3 4 5 9 1 2 3 4 5 9 1 2 3 5 9 2 4 6 9 1 2 4 5 6 9 ------------------------------------------------------------------------ 2 2 2 4 2 3 2 2 2 2 2 2x x x x x , x x x x - 4x x x x x - x x x x x + 4x x x x x x x + 1 2 5 6 9 3 4 5 8 1 3 4 5 9 2 3 4 6 9 1 2 3 4 5 6 9 ------------------------------------------------------------------------ 2 2 2 2 2 2 2 3 3 2 2 2 3 3x x x x x - 4x x x x x x + x x x x , x x x x x + x x x x - 1 3 5 6 9 1 2 3 5 6 9 1 2 6 9 1 3 4 5 8 2 3 4 9 ------------------------------------------------------------------------ 2 2 2 2 2 2 3 2 4x x x x x - 4x x x x x x + 4x x x x x x x + 3x x x x x + 1 3 4 5 9 1 2 3 4 6 9 1 2 3 4 5 6 9 1 3 5 6 9 ------------------------------------------------------------------------ 2 2 2 3 2 2 2 4 2 2 2 2 2 3 3x x x x x - 4x x x x x , x x x - x x x x - 4x x x x x + 1 2 4 6 9 1 2 5 6 9 2 3 7 1 3 7 8 1 2 3 7 9 ------------------------------------------------------------------------ 2 2 2 2 2 2 3 2 4 2 2 2 2 3 2 2 2 4x x x x x x + 3x x x x - 4x x x x x + x x x , x x x x - x x x x x 1 2 3 7 8 9 1 2 7 9 1 2 7 8 9 1 8 9 2 3 4 7 1 3 4 7 8 ------------------------------------------------------------------------ 2 2 2 2 2 2 3 2 - 4x x x x x x + 4x x x x x x x + 3x x x x x - 4x x x x x + 1 2 3 4 7 9 1 2 3 4 7 8 9 1 2 4 7 9 1 2 4 8 9 ------------------------------------------------------------------------ 4 2 2 2 2 2 2 2 2 2 2 2 2 2 x x x x , x x x x - x x x x - 4x x x x x x + 4x x x x x x + 1 5 8 9 2 3 4 7 1 3 4 8 1 2 3 4 7 9 1 2 3 4 8 9 ------------------------------------------------------------------------ 2 2 2 2 3 2 4 2 2 2 2 3 2 2 2 2 3 3x x x x - 4x x x x x + x x x , x x x x - x x x x x - 4x x x x x + 1 2 4 9 1 2 4 5 9 1 5 9 2 3 4 7 1 3 4 5 8 1 2 3 4 9 ------------------------------------------------------------------------ 2 2 2 2 2 3 4 2 2 3 4x x x x x x + 3x x x x x - 4x x x x x x + x x x x , x x x x x - 1 2 3 4 5 9 1 2 4 6 9 1 2 4 5 6 9 1 5 6 9 1 2 3 4 5 ------------------------------------------------------------------------ 2 2 4 3 2 2 2 2 3 4 2 2 3 2 x x x + x x x x x - 4x x x x x x + 3x x x x x - x x x + 3x x x x x 1 3 5 2 3 4 5 6 1 2 3 4 5 6 1 2 3 5 6 2 4 6 1 2 4 5 6 ------------------------------------------------------------------------ 2 2 2 2 4 2 2 3 2 2 2 2 2 2 2 2 2 2 2 - 2x x x x , x x x - 4x x x x x - x x x x + 4x x x x x x + 3x x x x 1 2 5 6 3 4 5 1 3 4 5 6 2 3 4 6 1 2 3 4 5 6 1 3 5 6 ------------------------------------------------------------------------ 2 3 2 2 4 3 2 2 2 2 3 2 2 2 2 2 2 - 4x x x x x + x x x , x x x x + x x x x - 4x x x x x - 4x x x x x 1 2 3 5 6 1 2 6 1 3 4 5 2 3 4 6 1 3 4 5 6 1 2 3 4 6 ------------------------------------------------------------------------ 2 2 3 2 2 2 2 3 3 3 2 2 4 2 2 2 2 + 4x x x x x x + 3x x x x + 3x x x x - 4x x x x , x x x - x x x x - 1 2 3 4 5 6 1 3 5 6 1 2 4 6 1 2 5 6 2 3 4 1 3 4 5 ------------------------------------------------------------------------ 2 3 2 2 2 2 2 2 3 2 4 2 2 4x x x x x + 4x x x x x x + 3x x x x - 4x x x x x + x x x ) 1 2 3 4 6 1 2 3 4 5 6 1 2 4 6 1 2 4 5 6 1 5 6 o5 : Ideal of R i6 : isPrimary Q o6 = true

## Ways to use joinIdeals :

• "joinIdeals(Ideal,Ideal)"

## For the programmer

The object joinIdeals is .