construct a regular system from a regular chain and a list of inequations
RegularSystem(rc, H, R)
list of polynomials of R
The command RegularSystem(rc, H, R) constructs a regular system from a regular chain and a list of inequations. Denote by W⁡T the quasi-component of rc. Then the constructed regular system encodes those points in W⁡T 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 1.
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.
Define a polynomial ring.
R ≔ PolynomialRing⁡x,y,z
Define a set of polynomials of R.
sys ≔ z⁢x2+y+z,y2+z
dec ≔ Triangularize⁡sys,R,output=lazard
There are two groups of solutions, each of which is given by a regular chain. To view the equations, use the Equations command.
Let rc1 be the first regular chain, and rc2 be the second one.
rc1,rc2 ≔ dec1,dec2
Consider two polynomials h1 and h2; regard them as inequations.
h1,h2 ≔ x,x+z
To obtain regular systems, first check if h1 is regular with respect to rc1, and h2 is regular with respect to rc2.
Both of them are regular, thus you can build the following regular systems.
rs1 ≔ RegularSystem⁡rc1,h1,R;rs2 ≔ RegularSystem⁡rc2,h2,R
You can simply call RegularSystem(R) to build the regular system which encodes all points.
ws ≔ ConstructibleSet⁡RegularSystem⁡R,R
The complement of ws must be empty.
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