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 $\mathrm{nops}\left(\mathrm{pt}\right)$ 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](..).

$\mathrm{with}\left(\mathrm{RegularChains}\right)\:$

$\mathrm{with}\left(\mathrm{ChainTools}\right)\:$

$\mathrm{with}\left(\mathrm{ConstructibleSetTools}\right)\:$

$\mathrm{with}\left(\mathrm{ParametricSystemTools}\right)\:$

$R≔\mathrm{PolynomialRing}\left(\left[x\,y\,s\right]\right)$

$R≔\mathrm{polynomial\_ring}$

$F≔\left[s-\left(y+1\right)x\,s-\left(x+1\right)y\right]$

$F≔\left[s-\left(y+1\right)x\,s-\left(x+1\right)y\right]$

$\mathrm{pctd},\mathrm{cells}≔\mathrm{ComprehensiveTriangularize}\left(F\,1\,R\right)$

$\mathrm{pctd},\mathrm{cells}≔\left[\mathrm{regular\_chain}\,\mathrm{regular\_chain}\,\mathrm{regular\_chain}\right],\left[\left[\mathrm{constructible\_set}\,\left[3\,2\right]\right]\,\left[\mathrm{constructible\_set}\,\left[1\right]\right]\right]$

$\mathrm{lcs}≔\left[\mathrm{seq}\left(\mathrm{cells}\left[i\right]\left[1\right]\,i=1..\mathrm{nops}\left(\mathrm{cells}\right)\right)\right]$

$\mathrm{lcs}≔\left[\mathrm{constructible\_set}\,\mathrm{constructible\_set}\right]$

$\mathrm{pt}≔\left[4\right]$

$\mathrm{pt}≔\left[4\right]$

$\mathrm{li}≔\mathrm{BelongsTo}\left(\mathrm{pt}\,\mathrm{lcs}\,R\right)\;$$i≔\mathrm{li}\left[1\right]$

$\mathrm{li}≔\left[2\right]$

$i≔2$

$\mathrm{ind}≔\mathrm{cells}\left[i\right]\left[2\right]$

$\mathrm{ind}≔\left[1\right]$

$\mathrm{lrc\_ind}≔\mathrm{map}\left(i\mapsto \mathrm{pctd}\left[i\right]\,\mathrm{ind}\right)$

$\mathrm{lrc\_ind}≔\left[\mathrm{regular\_chain}\right]$

$\mathrm{map}\left(\mathrm{Info}\,\mathrm{lrc\_ind}\,R\right)$

$\left[\left[\left(y+1\right)x-s\,y^2+y-s\right]\right]$

$\mathrm{lrc\_sp}≔\mathrm{Specialize}\left(\mathrm{pt}\,\mathrm{lrc\_ind}\,R\right)$

$\mathrm{lrc\_sp}≔\left[\mathrm{regular\_chain}\right]$

$\mathrm{map}\left(\mathrm{Info}\,\mathrm{lrc\_sp}\,R\right)$

$\left[\left[\left(y+1\right)x-4\,y^2+y-4\right]\right]$