The command RegularSystem(rc, H, R) constructs a regular system from a regular chain and a list of inequations. Denote by $W\left(T\right)$ the quasi-component of rc. Then the constructed regular system encodes those points in $W\left(T\right)$ that do not cancel any polynomial in H.

Each polynomial in H must be regular with respect to the regular chain rc; otherwise an error is reported.

If rc is not specified, then rc is set to the empty regular chain.

If H is not specified, then H is set to $\left[1\right]$.

The command RegularSystem(R) constructs the regular system corresponding to the whole space.

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

See ConstructibleSetTools and RegularChains for the related mathematical concepts, in particular for the ideas of a constructible set, a regular system, and a regular chain.

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

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

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

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

$R≔\mathrm{polynomial\_ring}$

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

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

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

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

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

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

$\mathrm{rc1}\,\mathrm{rc2}≔{\mathrm{dec}}_1\,{\mathrm{dec}}_2$

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

$\mathrm{h1}\,\mathrm{h2}≔x\,x\+z$

$\mathrm{h1},\mathrm{h2}≔x,x+z$

$\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)$

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

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

$\mathrm{ws}≔\mathrm{ConstructibleSet}\left(\left[\mathrm{RegularSystem}\left(R\right)\right]\,R\right)$

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

$\mathrm{IsEmpty}\left(\mathrm{Complement}\left(\mathrm{ws}\,R\right)\,R\right)$

$\mathrm{true}$