# numericalImageSample -- samples general points on the image of a variety

## Synopsis

• Usage:
numericalImageSample(F, I, P, s)
numericalImageSample(F, I, s)
numericalImageSample(F, I)
• Inputs:
• F, a matrix, or list, or ring map, specifying a map
• I, an ideal, which is prime, specifying a source variety $V(I)$
• P, a list, of points on $F(V(I))$
• s, an integer, the number of points to sample in $F(V(I))$
• Optional inputs:
• Software => ..., default value M2engine, specify software for homotopy continuation
• Outputs:
• a list, of sample points on $F(V(I)))$

## Description

This method computes a list of sample points on the image of a variety numerically, by calling numericalSourceSample.

If the number of points $s$ is unspecified, then it is assumed that $s = 1$.

One can optionally provide an initial list of points $P$ on $F(V(I))$, which will then be completed to a list of $s$ points on $F(V(I))$.

The following example samples a point from the twisted cubic. We then independently verify that this point does lie on the twisted cubic.

 i1 : R = CC[s,t]; i2 : F = {s^3,s^2*t,s*t^2,t^3}; i3 : p = first numericalImageSample(F, ideal 0_R) o3 = p o3 : Point i4 : A = matrix{p#Coordinates_{0,1,2}, p#Coordinates_{1,2,3}}; 2 3 o4 : Matrix CC <--- CC 53 53 i5 : numericalNullity A == 2 o5 = true

Here is how to sample a point from the Grassmannian $Gr(2,4)$ of $P^1$'s in $P^3$, under its Pl&uuml;cker embedding in $P^5$. We take maximal minors of a $2 x 4$ matrix, whose row span gives a $P^1$ in $P^3$.

 i6 : R = CC[x_(1,1)..x_(2,4)]; i7 : F = (minors(2, genericMatrix(R, 2, 4)))_*; i8 : numericalImageSample(F, ideal 0_R) o8 = {{-.434457-.140153*ii, -.374655-.460763*ii, -.192528-.151059*ii, ------------------------------------------------------------------------ -.113152+.276111*ii, .222593+.353053*ii, .100773+.425004*ii}} o8 : List

## Ways to use numericalImageSample :

• "numericalImageSample(List,Ideal)"
• "numericalImageSample(List,Ideal,List,ZZ)"
• "numericalImageSample(List,Ideal,ZZ)"
• "numericalImageSample(Matrix,Ideal)"
• "numericalImageSample(Matrix,Ideal,List,ZZ)"
• "numericalImageSample(Matrix,Ideal,ZZ)"
• "numericalImageSample(RingMap,Ideal)"
• "numericalImageSample(RingMap,Ideal,List,ZZ)"
• "numericalImageSample(RingMap,Ideal,ZZ)"

## For the programmer

The object numericalImageSample is .