generalEquations -- random linear combinations of equations/generators

Synopsis

Usage:

L = generalEquations(k,F)

L = generalEquations(k,I)
Inputs:
- k, an integer,
- F, a list, a list of polynomials
- I, an ideal,
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 a method function.