The command BelongsTo(pt, lcs, R) returns the indices of constructible sets in lcs which contain the point pt.

pt is given by a list of rational numbers; $\mathrm{nops}\left(\mathrm{pt}\right)$ must be equal to the number of variables of R.

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

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

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

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

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

$R≔\mathrm{polynomial\_ring}$

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

$S≔\mathrm{polynomial\_ring}$

$\mathrm{MP}≔\left[x^2\,y^2\right]$

$\mathrm{MP}≔\left[x^2\,y^2\right]$

$F≔\left[s-1\,t-1\right]$

$F≔\left[s-1\,t-1\right]$

$\mathrm{cs1}≔\mathrm{PolynomialMapPreimage}\left(F\,\mathrm{MP}\,R\,S\right)$

$\mathrm{cs1}≔\mathrm{constructible\_set}$

$\mathrm{Info}\left(\mathrm{cs1}\,R\right)$

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

$\mathrm{pt}≔\left[\frac{1}{2}\,\frac{1}{3}\right]$

$\mathrm{pt}≔\left[\frac{1}{2}\,\frac{1}{3}\right]$

$\mathrm{cs2}≔\mathrm{GeneralConstruct}\left(\left[2x-1\,3y-1\right]\,\left[1\right]\,R\right)$

$\mathrm{cs2}≔\mathrm{constructible\_set}$

$\mathrm{BelongsTo}\left(\mathrm{pt}\,\left[\mathrm{cs1}\,\mathrm{cs2}\right]\,R\right)$

$\left[2\right]$