specialize a list of regular chains at a point
Specialize(pt, lrc, R)
point with coordinates in rational number field or a finite field
list of regular chains
The command Specialize(pt, lrc, R) returns a list of regular chains obtained from those of lrc by specialization at the point pt.
The point pt is given by a list of rational numbers or a list of elements in a finite field; moreover, the number of coordinates in pt must be less than or equal to the number of variables of R.
All polynomials in each regular chain of lrc are evaluated at the last nops⁡pt variables of R using the corresponding coordinates of pt.
Regular chains in lrc must specialize well at pt, otherwise an error message displays.
This command is part of the RegularChains[ParametricSystemTools] package, so it can be used in the form Specialize(..) 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][Specialize](..).
The following example shows how to analyze the output of a comprehensive triangular decomposition.
R ≔ PolynomialRing⁡x,y,s
F ≔ s−y+1⁢x,s−x+1⁢y
pctd,cells ≔ ComprehensiveTriangularize⁡F,1,R
The first part is a list of regular chains which form a pre-comprehensive triangular decomposition of F. The second part is a partition of the projection image of V(F) to the last coordinate. Each constructible set is associated with indices of regular chains in the first part.
lcs ≔ seq⁡cellsi1,i=1..nops⁡cells
Consider a specialization point pt⁡s=4.
pt ≔ 4
Try to figure out to which partition pt belongs.
li ≔ BelongsTo⁡pt,lcs,R;i ≔ li1
Then retrieve the indices of regular chains that specialize well at pt.
ind ≔ cellsi2
lrc_ind ≔ map⁡i→pctdi,ind
Thus you know that the regular chains in lrc_ind all specialize well at the point pt. Then you can do simple substitutions.
lrc_sp ≔ Specialize⁡pt,lrc_ind,R
Regular chains of lrc_sp form a triangular decomposition of F after specialization at pt.
Download Help Document