The command PolynomialMapImage(F, PM, R, S) returns a constructible set cs which is the image of the variety $V\left(F\right)$ under the polynomial map PM.

The command PolynomialMapImage(F, H, PM, R, S) returns a constructible set cs which is the image of the difference of the variety $V\left(F\right)$ by the variety $V\left(H\right)$ under the polynomial map PM.

The command PolynomialMapImage(CS, PM, R, S) returns a constructible set cs which is the image of the constructible set CS under the polynomial map PM.

Both rings R and S should be over the same base field.

The variable sets of R and S should be disjoint.

The number of polynomials in PM is equal to the number of variables of ring S.

This command is part of the RegularChains[ConstructibleSetTools] package, so it can be used in the form PolynomialMapImage(..) only after executing the command with(RegularChains[ConstructibleSetTools]).  However, it can always be accessed through the long form of the command by using RegularChains[ConstructibleSetTools][PolynomialMapImage](..).

$\mathrm{with}\left(\mathrm{RegularChains}\right)\:$

$\mathrm{with}\left(\mathrm{ConstructibleSetTools}\right)\:$

$R≔\mathrm{PolynomialRing}\left(\left[u\,v\right]\right)$

$R≔\mathrm{polynomial\_ring}$

$S≔\mathrm{PolynomialRing}\left(\left[x\,y\,z\right]\right)$

$S≔\mathrm{polynomial\_ring}$

$\mathrm{PM}≔\left[uv\,u\,v^2\right]$

$\mathrm{PM}≔\left[uv\,u\,v^2\right]$

$\mathrm{cs}≔\mathrm{PolynomialMapImage}\left(\left[\right]\,\mathrm{PM}\,R\,S\right)$

$\mathrm{cs}≔\mathrm{constructible\_set}$

$\mathrm{cs}≔\mathrm{MakePairwiseDisjoint}\left(\mathrm{cs}\,S\right)$

$\mathrm{cs}≔\mathrm{constructible\_set}$

$\mathrm{Info}\left(\mathrm{cs}\,S\right)$

$\left[\left[x\,y\right]\,\left[1\right]\right],\left[\left[x^2-y^{2}z\right]\,\left[y\right]\right]$