return the list of inequations in a regular system
The command RepresentingInequations(rs, R) returns the inequations of the regular system rs, assuming that the polynomials of rs belong to R
This command is part of the RegularChains[ConstructibleSetTools] package, so it can be used in the form RepresentingInequations(..) 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][RepresentingInequations](..).
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 the polynomials system sys by means of regular chains.
dec ≔ Triangularize⁡sys,R,output=lazard
Let rc be the first regular chain and h be a polynomial regarded as an inequation.
rc ≔ dec1;h ≔ x+z
To obtain a regular system, check whether h is regular with respect to rc.
Since h is regular, you can build a regular system.
rs ≔ RegularSystem⁡rc,h,R
Notice that the inequation h is returned by the command RepresentingInequations.
ineqs ≔ RepresentingInequations⁡rs,R
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