return the list of regular systems in a constructible set
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](..).
First, define a polynomial ring R and two polynomials of R.
R ≔ PolynomialRing⁡x,y,u,v
f ≔ u⁢x+v;g ≔ v⁢y+u
Using GeneralConstruct, construct a constructible set from the common solutions of f and g which do not cancel u2+v2−1
cs ≔ GeneralConstruct⁡f,g,u2+v2−1,R
Now retrieve the regular systems from cs.
lrs ≔ RepresentingRegularSystems⁡cs,R
Next extract the representing chains and inequations
lrc ≔ map⁡RepresentingChain,lrs,R
The first inequation is u2+v2−1 since this polynomial can vanish inside the quasi-component of the first regular chain.
The second inequation is simply 1 since u2+v2−1 cannot vanish inside the quasi-component of the second regular chain.
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