The command PolynomialMapPreimage(F, PM, R, S) returns a constructible set cs over R, which is the preimage of the variety V(F) under the polynomial map PM.

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

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

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

The variable sets of R and S should be disjoint.

This command is part of the RegularChains[ConstructibleSetTools] package, so it can be used in the form PolynomialMapPreimage(..) 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][PolynomialMapPreimage](..).

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

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

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

$R≔\mathrm{polynomial\_ring}$

$S≔\mathrm{PolynomialRing}\left(\left[s\,t\right]\right)$

$S≔\mathrm{polynomial\_ring}$

$\mathrm{MP}≔\left[x^2\,y^2\right]$

$\mathrm{MP}≔\left[x^2\,y^2\right]$

$F≔\left[s-1\,t-1\right]$

$F≔\left[s-1\,t-1\right]$

$\mathrm{cs}≔\mathrm{PolynomialMapPreimage}\left(F\,\mathrm{MP}\,R\,S\right)$

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

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

$\left[\left[x+1\,y-1\right]\,\left[1\right]\right],\left[\left[x-1\,y-1\right]\,\left[1\right]\right],\left[\left[x+1\,y+1\right]\,\left[1\right]\right],\left[\left[x-1\,y+1\right]\,\left[1\right]\right]$