generalEquations -- random linear combinations of equations/generators

Synopsis

• Usage:
L = generalEquations(k,F)
L = generalEquations(k,I)
• Inputs:
• Outputs:
• L, a list, k linear combinations of polynomials in F (of generators of I)

Description

A variety V (that is not necessarily a complete intersection) of codimension k is a component of a complete intersection of codimension k defined by k general linear combinations of any generating set of the defining ideal of V.

This function automates the above construction.

 i1 : R = CC[x,y,z]; i2 : F = {x*y, x^2 - y, x*z}; i3 : L = generalEquations(2,F) 2 o3 = {(.29398 + .632944*ii)x + (.892712 + .673395*ii)x*y + (.0258884 + ------------------------------------------------------------------------ 2 .714827*ii)x*z + (- .29398 - .632944*ii)y, (.461944 + .775187*ii)x + ------------------------------------------------------------------------ (.89189 + .231053*ii)x*y + (.909047 + .314897*ii)x*z + (- .461944 - ------------------------------------------------------------------------ .775187*ii)y} o3 : List

Ways to use generalEquations :

• "generalEquations(WitnessSet)"
• "generalEquations(ZZ,Ideal)"
• "generalEquations(ZZ,List)"

For the programmer

The object generalEquations is .