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# generalEquations -- random linear combinations of equations/generators

- 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.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/NAGtypes/doc-NAGtypes.m2:830:0.