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RegularChains[ParametricSystemTools]

 Specialize
 specialize a list of regular chains at a point

 Calling Sequence Specialize(pt, lrc, R)

Parameters

 pt - point with coordinates in rational number field or a finite field lrc - list of regular chains R - polynomial ring

Description

 • 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](..).

Examples

 > $\mathrm{with}\left(\mathrm{RegularChains}\right):$
 > $\mathrm{with}\left(\mathrm{ChainTools}\right):$
 > $\mathrm{with}\left(\mathrm{ConstructibleSetTools}\right):$
 > $\mathrm{with}\left(\mathrm{ParametricSystemTools}\right):$

The following example shows how to analyze the output of a comprehensive triangular decomposition.

 > $R≔\mathrm{PolynomialRing}\left(\left[x,y,s\right]\right)$
 ${R}{≔}{\mathrm{polynomial_ring}}$ (1)
 > $F≔\left[s-\left(y+1\right)x,s-\left(x+1\right)y\right]$
 ${F}{≔}\left[{s}{-}\left({y}{+}{1}\right){}{x}{,}{s}{-}\left({x}{+}{1}\right){}{y}\right]$ (2)
 > $\mathrm{pctd},\mathrm{cells}≔\mathrm{ComprehensiveTriangularize}\left(F,1,R\right)$
 ${\mathrm{pctd}}{,}{\mathrm{cells}}{≔}\left[{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}\right]{,}\left[\left[{\mathrm{constructible_set}}{,}\left[{3}{,}{2}\right]\right]{,}\left[{\mathrm{constructible_set}}{,}\left[{1}\right]\right]\right]$ (3)

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.

 > $\mathrm{lcs}≔\left[\mathrm{seq}\left(\mathrm{cells}\left[i\right]\left[1\right],i=1..\mathrm{nops}\left(\mathrm{cells}\right)\right)\right]$
 ${\mathrm{lcs}}{≔}\left[{\mathrm{constructible_set}}{,}{\mathrm{constructible_set}}\right]$ (4)

Consider a specialization point $\mathrm{pt}\left(s=4\right)$.

 > $\mathrm{pt}≔\left[4\right]$
 ${\mathrm{pt}}{≔}\left[{4}\right]$ (5)

Try to figure out to which partition pt belongs.

 > $\mathrm{li}≔\mathrm{BelongsTo}\left(\mathrm{pt},\mathrm{lcs},R\right);$$i≔\mathrm{li}\left[1\right]$
 ${\mathrm{li}}{≔}\left[{2}\right]$
 ${i}{≔}{2}$ (6)

Then retrieve the indices of regular chains that specialize well at pt.

 > $\mathrm{ind}≔\mathrm{cells}\left[i\right]\left[2\right]$
 ${\mathrm{ind}}{≔}\left[{1}\right]$ (7)
 > $\mathrm{lrc_ind}≔\mathrm{map}\left(i↦\mathrm{pctd}\left[i\right],\mathrm{ind}\right)$
 ${\mathrm{lrc_ind}}{≔}\left[{\mathrm{regular_chain}}\right]$ (8)
 > $\mathrm{map}\left(\mathrm{Info},\mathrm{lrc_ind},R\right)$
 $\left[\left[\left({y}{+}{1}\right){}{x}{-}{s}{,}{{y}}^{{2}}{+}{y}{-}{s}\right]\right]$ (9)

Thus you know that the regular chains in lrc_ind all specialize well at the point pt. Then you can do simple substitutions.

 > $\mathrm{lrc_sp}≔\mathrm{Specialize}\left(\mathrm{pt},\mathrm{lrc_ind},R\right)$
 ${\mathrm{lrc_sp}}{≔}\left[{\mathrm{regular_chain}}\right]$ (10)

Regular chains of $\mathrm{lrc_sp}$ form a triangular decomposition of F after specialization at pt.

 > $\mathrm{map}\left(\mathrm{Info},\mathrm{lrc_sp},R\right)$
 $\left[\left[\left({y}{+}{1}\right){}{x}{-}{4}{,}{{y}}^{{2}}{+}{y}{-}{4}\right]\right]$ (11)