# point -- pick a random rational point on a projective variety

## Synopsis

• Usage:
point R
• Inputs:
• R, a 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, 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 0.446639 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 0.317576 seconds o3 = true