The command RegularSystemDifference(rs1, rs2, R) returns a constructible set which is the difference of rs1 and rs2.

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

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

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

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

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

$R≔\mathrm{polynomial\_ring}$

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

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

$\mathrm{dec}≔\mathrm{Triangularize}\left(\mathrm{sys}\,R\,\mathrm{output}=\mathrm{lazard}\right)$

$\mathrm{dec}≔\left[\mathrm{regular\_chain}\,\mathrm{regular\_chain}\right]$

$\mathrm{map}\left(\mathrm{Equations}\,\mathrm{dec}\,R\right)$

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

$\mathrm{rc1},\mathrm{rc2}≔\mathrm{dec}\left[1\right],\mathrm{dec}\left[2\right]$

$\mathrm{rc1},\mathrm{rc2}≔\mathrm{regular\_chain},\mathrm{regular\_chain}$

$\mathrm{h1},\mathrm{h2}≔x,x+z$

$\mathrm{h1},\mathrm{h2}≔x,x+z$

$\mathrm{IsRegular}\left(\mathrm{h1}\,\mathrm{rc1}\,R\right)\;$$\mathrm{IsRegular}\left(\mathrm{h2}\,\mathrm{rc2}\,R\right)$

$\mathrm{true}$

$\mathrm{true}$

$\mathrm{rs1}≔\mathrm{RegularSystem}\left(\mathrm{rc1}\,\left[\mathrm{h1}\right]\,R\right)\;$$\mathrm{rs2}≔\mathrm{RegularSystem}\left(\mathrm{rc2}\,\left[\mathrm{h2}\right]\,R\right)$

$\mathrm{rs1}≔\mathrm{regular\_system}$

$\mathrm{rs2}≔\mathrm{regular\_system}$

$\mathrm{cs}≔\mathrm{RegularSystemDifference}\left(\mathrm{rs1}\,\mathrm{rs2}\,R\right)$

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

$\mathrm{lrs}≔\mathrm{RepresentingRegularSystems}\left(\mathrm{cs}\,R\right)$

$\mathrm{lrs}≔\left[\mathrm{regular\_system}\right]$

$\mathrm{lrc}≔\mathrm{map}\left(\mathrm{RepresentingChain}\,\mathrm{lrs}\,R\right)$

$\mathrm{lrc}≔\left[\mathrm{regular\_chain}\right]$

$\mathrm{eqs}≔\mathrm{map}\left(\mathrm{Equations}\,\mathrm{lrc}\,R\right)$

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

$\mathrm{ineqs}≔\mathrm{map}\left(\mathrm{RepresentingInequations}\,\mathrm{lrs}\,R\right)$

$\mathrm{ineqs}≔\left[\left[x\,z\right]\right]$

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

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