The command RationalMapPreimage(F, RM, R, S) returns a constructible set cs over R. cs is the preimage of the variety $V\left(F\right)$ under the rational map RM.

If H is specified, let $W$ be the variety defined by the product of polynomials in H. The command RationalMapPreimage(F, H, RM, R, S) returns the preimage of the constructible set $V$-$W$ under the rational map RM.

The command RationalMapPreimage(CS, RM, R, S) returns the preimage of the constructible set CS under the rational map RM.

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

The variable sets of R and S should be disjoint.

The number of rational functions in RM is equal to the number of variables of ring S.

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

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

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

$R≔\mathrm{polynomial\_ring}$

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

$S≔\mathrm{polynomial\_ring}$

$\mathrm{RM}≔\left[\frac{x^2}{x+y}\,\frac{y^2}{x+y}\right]$

$\mathrm{RM}≔\left[\frac{x^2}{x+y}\,\frac{y^2}{x+y}\right]$

$F≔\left[s-1\,t-1\right]$

$F≔\left[s-1\,t-1\right]$

$\mathrm{cs}≔\mathrm{RationalMapPreimage}\left(F\,\mathrm{RM}\,R\,S\right)$

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

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

$\left[\left[x-2\,y-2\right]\,\left[1\right]\right]$