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  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](..).