RegularSystem - Maple Help

RegularChains[ConstructibleSetTools]

 RegularSystem
 construct a regular system from a regular chain and a list of inequations

 Calling Sequence RegularSystem(rc, H, R) RegularSystem(rc, R) RegularSystem(H, R) RegularSystem(R)

Parameters

 rc - regular chain H - list of polynomials of R R - polynomial ring

Description

 • 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.

Examples

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

Define a polynomial ring.

 > $R≔\mathrm{PolynomialRing}\left(\left[x,y,z\right]\right)$
 ${R}{≔}{\mathrm{polynomial_ring}}$ (1)

Define a set of polynomials of R.

 > $\mathrm{sys}≔\left[z{x}^{2}+y+z,{y}^{2}+z\right]$
 ${\mathrm{sys}}{≔}\left[{z}{}{{x}}^{{2}}{+}{y}{+}{z}{,}{{y}}^{{2}}{+}{z}\right]$ (2)
 > $\mathrm{dec}≔\mathrm{Triangularize}\left(\mathrm{sys},R,\mathrm{output}=\mathrm{lazard}\right)$
 ${\mathrm{dec}}{≔}\left[{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}\right]$ (3)

There are two groups of solutions, each of which is given by a regular chain. To view the equations, use the Equations command.

 > $\mathrm{map}\left(\mathrm{Equations},\mathrm{dec},R\right)$
 $\left[\left[{z}{}{{x}}^{{2}}{+}{y}{+}{z}{,}{{y}}^{{2}}{+}{z}\right]{,}\left[{y}{,}{z}\right]\right]$ (4)

Let rc1 be the first regular chain, and rc2 be the second one.

 > $\mathrm{rc1},\mathrm{rc2}≔\mathrm{dec}\left[1\right],\mathrm{dec}\left[2\right]$
 ${\mathrm{rc1}}{,}{\mathrm{rc2}}{≔}{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}$ (5)

Consider two polynomials h1 and h2; regard them as inequations.

 > $\mathrm{h1},\mathrm{h2}≔x,x+z$
 ${\mathrm{h1}}{,}{\mathrm{h2}}{≔}{x}{,}{x}{+}{z}$ (6)

To obtain regular systems, first check if $\mathrm{h1}$ is regular with respect to $\mathrm{rc1}$, and $\mathrm{h2}$ is regular with respect to $\mathrm{rc2}$.

 > $\mathrm{IsRegular}\left(\mathrm{h1},\mathrm{rc1},R\right);$$\mathrm{IsRegular}\left(\mathrm{h2},\mathrm{rc2},R\right)$
 ${\mathrm{true}}$
 ${\mathrm{true}}$ (7)

Both of them are regular, thus you can build the following regular systems.

 > $\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)$
 ${\mathrm{rs1}}{≔}{\mathrm{regular_system}}$
 ${\mathrm{rs2}}{≔}{\mathrm{regular_system}}$ (8)

You can simply call RegularSystem(R) to build the regular system which encodes all points.

 > $\mathrm{ws}≔\mathrm{ConstructibleSet}\left(\left[\mathrm{RegularSystem}\left(R\right)\right],R\right)$
 ${\mathrm{ws}}{≔}{\mathrm{constructible_set}}$ (9)

The complement of $\mathrm{ws}$ must be empty.

 > $\mathrm{IsEmpty}\left(\mathrm{Complement}\left(\mathrm{ws},R\right),R\right)$
 ${\mathrm{true}}$ (10)