findPoint -- Find a kk-rational point in a variety

Synopsis

Usage:

point=findPoint c
Inputs:
- I, an ideal,
Optional inputs:
- Verbose => ..., default value false
Outputs:
- B, a matrix, coordinates of a point in the finite ground field

Description

Given ideal c the functions adds random linear equations L to c to obtain 1-dimensional ideal. Since the ground field is finite, decompose the ideal c+L will lead to a point with positive probability. Thus repeating will lead to success.

i1 : kk=ZZ/101

o1 = kk

o1 : QuotientRing

i2 : R=kk[x_0..x_6]

o2 = R

o2 : PolynomialRing

i3 : c=ideal random(R^1,R^{2:-1,2:-2})

                                                                          
o3 = ideal (24x  - 30x  + 19x  - 10x  - 8x  - 29x  - 38x , - 36x  - 29x  +
               0      1      2      3     4      5      6       0      1  
     ------------------------------------------------------------------------
                                          2               2                  
     19x  - 29x  - 22x  - 24x  - 16x , 39x  + 34x x  + 16x  - 47x x  + 45x x 
        2      3      4      5      6     0      0 1      1      0 2      1 2
     ------------------------------------------------------------------------
          2                                 2                             
     + 39x  - 18x x  - 48x x  - 17x x  + 40x  - 43x x  + 47x x  + 48x x  +
          2      0 3      1 3      2 3      3      0 4      1 4      2 4  
     ------------------------------------------------------------------------
                 2                                                2         
     46x x  - 23x  - 28x x  - 16x x  + 35x x  + x x  + 2x x  - 37x  + 38x x 
        3 4      4      0 5      1 5      2 5    3 5     4 5      5      0 6
     ------------------------------------------------------------------------
                                                       2     2           
     + 15x x  - 38x x  + 22x x  - 47x x  - 10x x  - 18x , 21x  + 19x x  +
          1 6      2 6      3 6      4 6      5 6      6     0      0 1  
     ------------------------------------------------------------------------
        2                        2                                 2         
     22x  - 39x x  - 34x x  + 43x  - 13x x  - 47x x  - 11x x  + 11x  - 15x x 
        1      0 2      1 2      2      0 3      1 3      2 3      3      0 4
     ------------------------------------------------------------------------
                                    2                                    
     + 19x x  + 36x x  - 28x x  - 7x  - 47x x  + 7x x  + 11x x  - 3x x  +
          1 4      2 4      3 4     4      0 5     1 5      2 5     3 5  
     ------------------------------------------------------------------------
                 2                                                       
     29x x  - 13x  + 2x x  - 23x x  + 33x x  - 47x x  + 15x x  + 30x x  +
        4 5      5     0 6      1 6      2 6      3 6      4 6      5 6  
     ------------------------------------------------------------------------
        2
     39x )
        6

o3 : Ideal of R

i4 : B=findPoint c

o4 = | -43 -32 24 -21 -15 3 -34 |

             1      7
o4 : Matrix R  <-- R

i5 : sub(c,B)==0

o5 = true

Ways to use findPoint:

findPoint(Ideal)

For the programmer

The object findPoint is a method function with options.