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RegularChains[ParametricSystemTools][Specialize] - specialize a list of regular chains at a point
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Calling Sequence
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Specialize(pt, lrc, R)
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Parameters
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pt
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point with coordinates in rational number field or a finite field
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lrc
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list of regular chains
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R
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polynomial ring
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Description
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The command Specialize(pt, lrc, R) returns a list of regular chains obtained from those of lrc by specialization at the point pt.
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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.
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All polynomials in each regular chain of lrc are evaluated at the last  variables of R using the corresponding coordinates of pt.
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Regular chains in lrc must specialize well at pt, otherwise an error message displays.
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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](..).
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Examples
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The following example shows how to analyze the output of a comprehensive triangular decomposition.
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![R := PolynomialRing([x, y, s])](/support/helpjp/helpview.aspx?si=6425/file06519/math125.png)
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![F := [s-(y+1)*x, s-(x+1)*y]](/support/helpjp/helpview.aspx?si=6425/file06519/math132.png)
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![F := [s-(y+1)*x, s-(x+1)*y]](/support/helpjp/helpview.aspx?si=6425/file06519/math135.png)
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![pctd, cells := [regular_chain, regular_chain, regular_chain], [[constructible_set, [3, 2]], [constructible_set, [1]]]](/support/helpjp/helpview.aspx?si=6425/file06519/math142.png)
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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.
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![lcs := [seq(cells[i][1], i = 1 .. nops(cells))]](/support/helpjp/helpview.aspx?si=6425/file06519/math152.png)
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![lcs := [constructible_set, constructible_set]](/support/helpjp/helpview.aspx?si=6425/file06519/math155.png)
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Consider a specialization point  .
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![pt := [4]](/support/helpjp/helpview.aspx?si=6425/file06519/math163.png)
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![pt := [4]](/support/helpjp/helpview.aspx?si=6425/file06519/math166.png)
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Try to figure out to which partition pt belongs.
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 ![i := li[1]](/support/helpjp/helpview.aspx?si=6425/file06519/math175.png)
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Then retrieve the indices of regular chains that specialize well at pt.
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![ind := cells[i][2]](/support/helpjp/helpview.aspx?si=6425/file06519/math189.png)
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![ind := [1]](/support/helpjp/helpview.aspx?si=6425/file06519/math192.png)
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![lrc_ind := map(proc (i) options operator, arrow; pctd[i] end proc, ind)](/support/helpjp/helpview.aspx?si=6425/file06519/math196.png)
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![lrc_ind := [regular_chain]](/support/helpjp/helpview.aspx?si=6425/file06519/math199.png)
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![[[(y+1)*x-s, y^2+y-s]]](/support/helpjp/helpview.aspx?si=6425/file06519/math206.png)
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Thus you know that the regular chains in lrc_ind all specialize well at the point pt. Then you can do simple substitutions.
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![lrc_sp := [regular_chain]](/support/helpjp/helpview.aspx?si=6425/file06519/math219.png)
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Regular chains of  form a triangular decomposition of F after specialization at pt.
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![[[(y+1)*x-4, y^2+y-4]]](/support/helpjp/helpview.aspx?si=6425/file06519/math234.png)
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