RepresentingRegularSystems - Maple Help

RegularChains[ConstructibleSetTools]

 RepresentingRegularSystems
 return the list of regular systems in a constructible set

 Calling Sequence RepresentingRegularSystems(cs, R)

Parameters

 cs - constructible set R - polynomial ring

Description

 • The command RepresentingRegularSystems(cs,R) returns a list of regular systems which defines the constructible set cs, that is, a list of regular systems (whose polynomials belong to R) such that the union of their zero sets is exactly equal to cs.
 • Recall that every constructible set built by the ConstructibleSetTools module is in fact represented by a list of regular systems representing it in the above sense.
 • 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.
 • The command RepresentingRegularSystems is part of the RegularChains[ConstructibleSetTools] package, so it can be used in the form RepresentingRegularSystems(..) 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][RepresentingRegularSystems](..).

Examples

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

First, define a polynomial ring $R$ and two polynomials of $R$.

 > $R≔\mathrm{PolynomialRing}\left(\left[x,y,u,v\right]\right)$
 ${R}{≔}{\mathrm{polynomial_ring}}$ (1)
 > $f≔ux+v;$$g≔vy+u$
 ${f}{≔}{u}{}{x}{+}{v}$
 ${g}{≔}{v}{}{y}{+}{u}$ (2)

Using GeneralConstruct, construct a constructible set from the common solutions of $f$ and $g$ which do not cancel ${u}^{2}+{v}^{2}-1$

 > $\mathrm{cs}≔\mathrm{GeneralConstruct}\left(\left[f,g\right],\left[{u}^{2}+{v}^{2}-1\right],R\right)$
 ${\mathrm{cs}}{≔}{\mathrm{constructible_set}}$ (3)

Now retrieve the regular systems from cs.

 > $\mathrm{lrs}≔\mathrm{RepresentingRegularSystems}\left(\mathrm{cs},R\right)$
 ${\mathrm{lrs}}{≔}\left[{\mathrm{regular_system}}{,}{\mathrm{regular_system}}\right]$ (4)

Next extract the representing chains and inequations

 > $\mathrm{lrc}≔\mathrm{map}\left(\mathrm{RepresentingChain},\mathrm{lrs},R\right)$
 ${\mathrm{lrc}}{≔}\left[{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}\right]$ (5)
 > $\mathrm{map}\left(\mathrm{Equations},\mathrm{lrc},R\right)$
 $\left[\left[{u}{}{x}{+}{v}{,}{v}{}{y}{+}{u}\right]{,}\left[{u}{,}{v}\right]\right]$ (6)
 > $\mathrm{map}\left(\mathrm{RepresentingInequations},\mathrm{lrs},R\right)$
 $\left[\left[{{u}}^{{2}}{+}{{v}}^{{2}}{-}{1}\right]{,}\left[\right]\right]$ (7)

The first inequation is ${u}^{2}+{v}^{2}-1$ since this polynomial can vanish inside the quasi-component of the first regular chain.

The second inequation is simply $1$ since ${u}^{2}+{v}^{2}-1$ cannot vanish inside the quasi-component of the second regular chain.