QuasiComponent - Maple Help

RegularChains[ConstructibleSetTools]

 QuasiComponent
 construct a constructible set from a regular chain

 Calling Sequence QuasiComponent(rc, R)

Parameters

 rc - regular chain R - polynomial ring

Description

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{RegularChains}\right):$
 > $\mathrm{with}\left(\mathrm{ConstructibleSetTools}\right):$
 > $R≔\mathrm{PolynomialRing}\left(\left[x,y,u,v\right]\right)$
 ${R}{≔}{\mathrm{polynomial_ring}}$ (1)
 > $F≔\left[ux+v,vy+u\right]$
 ${F}{≔}\left[{u}{}{x}{+}{v}{,}{v}{}{y}{+}{u}\right]$ (2)
 > $\mathrm{dec}≔\mathrm{Triangularize}\left(F,R,\mathrm{output}=\mathrm{lazard}\right)$
 ${\mathrm{dec}}{≔}\left[{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}\right]$ (3)
 > $\mathrm{map}\left(\mathrm{Equations},\mathrm{dec},R\right)$
 $\left[\left[{u}{}{x}{+}{v}{,}{v}{}{y}{+}{u}\right]{,}\left[{u}{,}{v}\right]\right]$ (4)
 > $\mathrm{map}\left(\mathrm{Inequations},\mathrm{dec},R\right)$
 $\left[\left\{{u}{,}{v}\right\}{,}{\varnothing }\right]$ (5)
 > $\mathrm{cs1}≔\mathrm{QuasiComponent}\left(\mathrm{dec}\left[1\right],R\right);$$\mathrm{cs2}≔\mathrm{QuasiComponent}\left(\mathrm{dec}\left[2\right],R\right)$
 ${\mathrm{cs1}}{≔}{\mathrm{constructible_set}}$
 ${\mathrm{cs2}}{≔}{\mathrm{constructible_set}}$ (6)
 > $\mathrm{Info}\left(\mathrm{cs1},R\right);$$\mathrm{Info}\left(\mathrm{cs2},R\right)$
 $\left[\left[{u}{}{x}{+}{v}{,}{v}{}{y}{+}{u}\right]{,}\left[{1}\right]\right]$
 $\left[\left[{u}{,}{v}\right]{,}\left[{1}\right]\right]$ (7)
 > $\left\{\mathrm{Info}\left(\mathrm{Union}\left(\mathrm{cs1},\mathrm{cs2},R\right),R\right)\right\}$
 $\left\{\left[\left[{u}{,}{v}\right]{,}\left[{1}\right]\right]{,}\left[\left[{u}{}{x}{+}{v}{,}{v}{}{y}{+}{u}\right]{,}\left[{1}\right]\right]\right\}$ (8)