compute the defining set of a regular chain
DefiningSet(rc, d, R)
number of parameters
The command DefiningSet(rc, d, R) returns the defining set of rc with respect to the last d variables, regarded as parameters. This is a constructible set C.
Given a positive integer d, the regular chain rc can be split into two parts. Denote by rc0 the set of the polynomials in rc involving only the last d variables, and denote by rc1 the other polynomials of rc. Certainly, both rc0 and rc1 are regular chains.
Let W be the quasi-component of rc0. For a point P in W, after specializing rc1 at P, two situations arise:
(1) either rc1 is not a regular chain anymore;
(2) or rc1 is still a regular chain.
There is a subtle point: after specializing rc1 at P, it might happen that it is still a regular chain, but its shape changes. In other words, the degree of the geometric object given by rc1 could change. The term specialize well, defined below, takes these cases into account.
The regular chain rc specializes well at a point P of W if rc1 is a regular chain after specialization and no initial of polynomials in rc1 vanish during the specialization.
The defining set of rc with respect to the last d variables consists of the points in W at which rc specializes well.
This command is part of the RegularChains[ParametricSystemTools] package, so it can be used in the form DefiningSet(..) only after executing the command with(RegularChains[ParametricSystemTools]). However, it can always be accessed through the long form of the command by using RegularChains[ParametricSystemTools][DefiningSet](..).
R ≔ PolynomialRing⁡x,y,u,v
Consider the following parametric polynomial system F.
F ≔ v⁢x⁢y+u⁢x2+x,u⁢y2+x2
For different values of u and v, the solution set has a different nature. For example, u=0 and v=0 is a degenerate case: x=0 and y can be any value. To understand more about F, first decompose F into a set of regular chains.
dec ≔ Triangularize⁡F,R,output=lazard
The first regular chain is simple. For all values of u and v, it is well-specialized.
ds1 ≔ DefiningSet⁡dec1,2,R;Info⁡ds1,R
For the last one, its defining set is given by u3+v2=0 and v≠0, and the inequality is to ensure that rc1 specializes well.
ds4 ≔ DefiningSet⁡dec4,2,R;Info⁡ds4,R
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