RepresentingChain - Maple Help

RegularChains[ConstructibleSetTools]

 RepresentingChain
 return the regular chain in a regular system

 Calling Sequence RepresentingChain(rs, R)

Parameters

 rs - regular system R - polynomial ring

Description

 • The command RepresentingChain(rs, R) returns the representing regular chain of the regular system rs, where the polynomials of rs belong to R.
 • This command is part of the RegularChains[ConstructibleSetTools] package, so it can be used in the form RepresentingChain(..) 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][RepresentingChain](..).
 • 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)

The command Triangularize (with lazard option) will decompose the common solutions of polynomial system $\mathrm{sys}$ using regular chains.

 > $\mathrm{dec}≔\mathrm{Triangularize}\left(\mathrm{sys},R,\mathrm{output}=\mathrm{lazard}\right)$
 ${\mathrm{dec}}{≔}\left[{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}\right]$ (3)

Let $\mathrm{rc}$ be the first regular chain.

 > $\mathrm{rc}≔\mathrm{dec}\left[1\right]$
 ${\mathrm{rc}}{≔}{\mathrm{regular_chain}}$ (4)

Consider a polynomial $h$, and regard it as an inequation.

 > $h≔x+z$
 ${h}{≔}{x}{+}{z}$ (5)

To obtain a regular system, check if $h$ is regular with respect to $\mathrm{rc}$.

 > $\mathrm{IsRegular}\left(h,\mathrm{rc},R\right)$
 ${\mathrm{true}}$ (6)

Since $h$ is regular, you can build a regular system.

 > $\mathrm{rs}≔\mathrm{RegularSystem}\left(\mathrm{rc},\left[h\right],R\right)$
 ${\mathrm{rs}}{≔}{\mathrm{regular_system}}$ (7)

Retrieve the regular chain by using the command RepresentingChain.

 > $\mathrm{EqualSaturatedIdeals}\left(\mathrm{rc},\mathrm{RepresentingChain}\left(\mathrm{rs},R\right),R\right)$
 ${\mathrm{true}}$ (8)