The command QuasiComponent(rc, R) returns a constructible set cs that encodes the quasi-component of the regular chain rc, that is, those points that cancel all equations of rc, but don't cancel any of the initials of the polynomials in rc.

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

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.

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

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

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

$R≔\mathrm{polynomial\_ring}$

$F≔\left[ux+v\,vy+u\right]$

$F≔\left[ux+v\,vy+u\right]$

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

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

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

$\left[\left[ux+v\,vy+u\right]\,\left[u\,v\right]\right]$

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

$\left[\left\{u\,v\right\}\,\varnothing \right]$

$\mathrm{cs1}≔\mathrm{QuasiComponent}\left(\mathrm{dec}\left[1\right]\,R\right)\;$$\mathrm{cs2}≔\mathrm{QuasiComponent}\left(\mathrm{dec}\left[2\right]\,R\right)$

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

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

$\mathrm{Info}\left(\mathrm{cs1}\,R\right)\;$$\mathrm{Info}\left(\mathrm{cs2}\,R\right)$

$\left[\left[ux+v\,vy+u\right]\,\left[1\right]\right]$

$\left[\left[u\,v\right]\,\left[1\right]\right]$

$\left\{\mathrm{Info}\left(\mathrm{Union}\left(\mathrm{cs1}\,\mathrm{cs2}\,R\right)\,R\right)\right\}$

$\left\{\left[\left[u\,v\right]\,\left[1\right]\right]\,\left[\left[ux+v\,vy+u\right]\,\left[1\right]\right]\right\}$