point(QuotientRing) -- pick a random rational point on a projective variety

Synopsis

Function: point
Usage:

point R
Inputs:
- R, a quotient ring, the homogeneous coordinate ring of a closed subscheme $X\subseteq\mathbb{P}^n$ over a finite ground field
Outputs:
- an ideal, an ideal in R defining a point on $X$

Description

This function is a variant of the randomKRationalPoint function, which has been further improved and extended in the package MultiprojectiveVarieties (missing documentation) , see point(MultiprojectiveVariety).

Below we verify the birationality of a rational map.

i1 : f = inverseMap specialQuadraticTransformation(9,ZZ/33331);

o1 : RationalMap (cubic rational map from 8-dimensional subvariety of PP^11 to PP^8)

i2 : time p = point source f
     -- used 0.109625 seconds

o2 = ideal (y   - 9235y  , y  + 11075y  , y  - 5847y  , y  + 7396y  , y  +
             10        11   9         11   8        11   7        11   6  
     ------------------------------------------------------------------------
     13530y  , y  + 4359y  , y  - 2924y  , y  + 13040y  , y  + 6904y  , y  -
           11   5        11   4        11   3         11   2        11   1  
     ------------------------------------------------------------------------
     12227y  , y  - 5653y  )
           11   0        11

                                                             ZZ
                                                           -----[y ..y  ]
                                                           33331  0   11
o2 : Ideal of -------------------------------------------------------------------------------------------------------
              (y y  - y y  + y y  , y y  - y y  + y y  , y y  - y y  + y y  , y y  - y y  + y y , y y  - y y  + y y )
                6 7    5 8    4 11   3 7    2 8    1 11   3 5    2 6    0 11   3 4    1 6    0 8   2 4    1 5    0 7

i3 : time p == f^* f p
     -- used 0.0907121 seconds

o3 = true

Ways to use this method: