# isOnImage -- tests whether a point lies on the image of a variety

## Synopsis

• Usage:
isOnImage(W, p)
isOnImage(F, I, p)
• Inputs:
• W, , a pseudo-witness set for $F(V(I))$
• p, , a point in the ambient space of $F(V(I))$
• F, a matrix, or list, or ring map, specifying a map
• I, an ideal, which is prime, specifying a source variety $V(I)$
• Optional inputs:
• MaxThreads => ..., default value 1, specifies the maximum number of processor threads
• Software => ..., default value M2engine, specify software for homotopy continuation
• Threshold => ..., default value 5, computes a pseudo-witness set for the image of a variety
• Verbose => ..., default value true, display detailed output
• Outputs:
• , whether the point $p$ lies on $F(V(I))$

## Description

This method determines if a point in the ambient target space lies on the image of a variety. This is done via computing a pseudo-witness set for the image.

If a pseudo-witness set has already been computed, then to avoid repetitive calculation one may run this function with the pseudo-witness set as input.

The following example determines whether a point lies on the Grassmannian $Gr(2,4)$ of $P^1$'s in $P^3$, under its Pl&uuml;cker embedding in $P^5$.

 i1 : R = CC[x_(1,1)..x_(2,4)]; I = ideal 0_R; o2 : Ideal of R i3 : F = (minors(2, genericMatrix(R, 2, 4)))_*; i4 : W = pseudoWitnessSet(F, I, Repeats => 2, Verbose => false); i5 : q = first numericalImageSample(F, I) o5 = q o5 : Point i6 : isOnImage(W, q) o6 = true i7 : isOnImage(W, point random(CC^1, CC^#F)) o7 = false i8 : isOnImage(W, point{{1_CC,0,0,0,0,0}}) o8 = true

## Ways to use isOnImage :

• "isOnImage(List,Ideal,Point)"
• "isOnImage(Matrix,Ideal,Point)"
• "isOnImage(PseudoWitnessSet,Point)"
• "isOnImage(RingMap,Ideal,Point)"

## For the programmer

The object isOnImage is .