# kernel(RingMap,ZZ) -- homogeneous components of the kernel of a homogeneous ring map

## Synopsis

• Function: kernel
• Usage:
kernel(phi,d)
• Inputs:
• phi, , $K[y_0,\ldots,y_m]/J \to K[x_0,\ldots,x_n]/I$, defined by homogeneous forms of the same degree and where $J$ and $I$ are homogeneous ideals
• d, an integer
• Optional inputs:
• Outputs:
• the ideal generated by all homogeneous elements of degree d belonging to the kernel of phi

## Description

This is equivalent to ideal image basis(d,kernel phi), but we use a more direct algorithm. We take advantage of the homogeneity and reduce the problem to linear algebra. For small values of d this method can be very fast, as the following example shows.

 i1 : phi = toMap map specialQuadraticTransformation 8 2 2 2 2 2 2 2 2 2 2 2 2 2 2 o1 = map (QQ[x ..x ], QQ[y ..y ], {- 5x x + x x + x x + 35x x - 7x x + x x - x x - 49x - 5x x + 2x x - x x + 27x x - 4x + x x - 7x x + 2x x - 2x x + 14x x - 4x x , - x x - 6x x - 5x x + 2x x + x x + x x - 5x x - x x + 2x x + 7x x - 2x x + 2x x - 3x x , - 25x + 9x x + 10x x - 2x x - x + 29x x - x x - 7x x - 13x x + 3x x + x x - x x + 2x x - x x + 7x x - 2x x - 8x x + 2x x - 3x x , x x + x x + x + 7x x - 9x x + 12x x - 4x + 2x x + 2x x - 14x x + 4x x + x x - x x - 14x x + x x , - 5x x + x x - 7x x + 8x x - 5x x + 2x x - x x + x x - x x + 7x x - 2x x - x x + 7x x - 2x x , x x + x - 7x x - 8x x + x x + x x + 2x x - x x + x x - 7x x + 2x x + x x - 7x x + 2x x , x x + x - 8x x + x x + 6x x - 2x + x x + x x - 7x x + 2x x + x x - 7x x + 2x x , x x - 7x x + x x + x x - 7x x + 2x - x x , - 4x x + x x - x - 7x x + 8x x + x x - x x - 6x x + 2x - x x - x x + 7x x - 2x x - x x + 7x x - 2x x , - 5x x + 2x + x x - x - x x + 8x x - 10x x + 2x x + 2x x - 2x x + 14x x - 4x x + 5x x - 3x x - 2x x + 7x x - 2x x - 3x x , - 5x x + x x + x x - 4x x - x x + x x + x x , x x - x x + 5x x + x x - 14x x - x x - 8x x - 8x x + 2x x + 4x x + 2x x + 4x x + 3x x - 7x x + 2x x - 3x x }) 0 8 0 11 0 3 2 4 3 4 0 5 2 5 3 5 4 5 5 0 6 2 6 4 6 5 6 6 4 7 5 7 6 7 4 8 5 8 6 8 1 2 1 5 0 6 1 6 4 6 5 6 0 7 1 7 2 7 5 7 6 7 1 8 7 8 0 0 2 0 4 2 4 4 0 5 2 5 4 5 0 6 4 6 5 6 0 7 2 7 4 7 5 7 6 7 0 8 4 8 7 8 2 4 3 4 4 2 5 4 5 5 6 6 3 7 4 7 5 7 6 7 3 8 4 8 5 8 6 8 0 4 2 4 2 5 4 5 0 6 2 6 4 6 5 6 4 7 5 7 6 7 4 8 5 8 6 8 0 4 4 1 5 4 5 0 6 1 6 4 6 5 6 4 7 5 7 6 7 4 8 5 8 6 8 2 3 4 4 5 4 6 5 6 6 3 7 4 7 5 7 6 7 4 8 5 8 6 8 1 3 1 5 1 6 4 6 5 6 6 3 7 0 3 3 4 4 0 5 4 5 0 6 4 6 5 6 6 3 7 4 7 5 7 6 7 4 8 5 8 6 8 0 2 2 2 4 4 2 5 4 5 0 6 5 6 2 7 4 7 5 7 6 7 0 8 2 8 4 8 5 8 6 8 7 8 0 1 1 2 1 4 0 6 1 6 4 6 0 7 0 2 1 2 0 4 1 4 1 5 2 5 4 5 0 6 1 6 4 6 2 7 0 8 1 8 5 8 6 8 7 8 o1 : RingMap QQ[x ..x ] <--- QQ[y ..y ] 0 8 0 11 i2 : time kernel(phi,1) -- used 0.0771615 seconds o2 = ideal () o2 : Ideal of QQ[y ..y ] 0 11 i3 : time kernel(phi,2) -- used 1.88555 seconds 2 o3 = ideal (y y + y y + y + 5y y + y y + 5y y - y y - 4y y - 5y y - 2 4 3 4 4 2 5 3 5 4 5 1 6 2 6 5 6 ------------------------------------------------------------------------ 4y y - 2y y - y y + 4y y - 5y y - 4y y + 3y y - 4y y - y y - 2 7 4 7 1 8 4 8 5 8 5 9 7 9 8 9 3 10 ------------------------------------------------------------------------ 2 3y y - 5y y - y y + 4y y + 5y y , 3y y - y y - 3y y - y + 6 10 8 10 4 11 6 11 8 11 1 3 2 3 3 4 4 ------------------------------------------------------------------------ 2y y - y y + y y + 2y y + 3y y - 7y y - 4y y + 7y y - 2y y + 0 5 3 5 1 6 2 6 5 6 2 7 4 7 1 8 4 8 ------------------------------------------------------------------------ y y - y y + 2y y + 2y y + y y - 7y y + 5y y - 3y y - y y - 0 9 4 9 5 9 7 9 8 9 0 10 3 10 6 10 0 11 ------------------------------------------------------------------------ 2y y - 2y y , 7y y + y y + 7y y - y y + 8y y - y y - y y + 3 11 4 11 0 1 0 4 1 4 3 4 0 5 3 5 1 6 ------------------------------------------------------------------------ 7y y + 8y y + y y + 8y y - y y - 8y y + 7y y - 8y y + 7y y + 2 6 5 6 2 7 4 7 1 8 4 8 5 9 7 9 8 9 ------------------------------------------------------------------------ y y - y y + 8y y - 7y y - 7y y - 7y y ) 0 10 3 10 6 10 0 11 4 11 6 11 o3 : Ideal of QQ[y ..y ] 0 11