return the regular chain in a regular system
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.
Define a polynomial ring.
R ≔ PolynomialRing⁡x,y,z
Define a set of polynomials of R.
sys ≔ z⁢x2+y+z,y2+z
The command Triangularize (with lazard option) will decompose the common solutions of polynomial system sys using regular chains.
dec ≔ Triangularize⁡sys,R,output=lazard
Let rc be the first regular chain.
rc ≔ dec1
Consider a polynomial h, and regard it as an inequation.
h ≔ x+z
To obtain a regular system, check if h is regular with respect to rc.
Since h is regular, you can build a regular system.
rs ≔ RegularSystem⁡rc,h,R
Retrieve the regular chain by using the command RepresentingChain.
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