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RegularChains[ChainTools][Dimension] - dimension of a regular chain
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Calling Sequence
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Dimension(rc, R)
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Parameters
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rc
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regular chain of R
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R
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polynomial ring
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Description
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The command Dimension(rc, R) returns the dimension of the saturated ideal of rc. This is also the number of variables of R minus the number of elements in rc.
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This command is part of the RegularChains[ChainTools] package, so it can be used in the form Dimension(..) only after executing the command with(RegularChains[ChainTools]). However, it can always be accessed through the long form of the command by using RegularChains[ChainTools][Dimension](..).
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Examples
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![R := PolynomialRing([x, y, a, b, c, d, g, h])](/support/helpjp/helpview.aspx?si=6432/file06439/math83.png)
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![decl := [regular_chain, regular_chain, regular_chain, regular_chain, regular_chain, regular_chain, regular_chain, regular_chain, regular_chain, regular_chain, regular_chain]](/support/helpjp/helpview.aspx?si=6432/file06439/math100.png)
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![[[c*x+d*y-h, (d*a-b*c)*y-h*a+c*g], [c*x+d*y-h, d*a-b*c, h*b-d*g], [a*x+b*y-g, d*y-h, c], [d*y-h, a, h*b-d*g, c], [c*x-h, h*a-c*g, b, d], [a*x+b*y-g, c, d, h], [c*x+d*y, d*a-b*c, g, h], [b*y-g, a, c, d, h], [y, a, c, g, h], [x, b, d, g, h], [a, b, c, d, g, h]]](/support/helpjp/helpview.aspx?si=6432/file06439/math107.png)
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We see that RegularChains[Triangularize] produces the regular chains in decreasing order of dimension. This is, in fact, part of the specifications of this function.
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![[6, 5, 5, 4, 4, 4, 4, 3, 3, 3, 2]](/support/helpjp/helpview.aspx?si=6432/file06439/math119.png)
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Here is another simple example with a triangular decomposition containing regular chains of different dimensions.
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![R := PolynomialRing([x, y, z])](/support/helpjp/helpview.aspx?si=6432/file06439/math125.png)
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| (6) |
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![dec := [regular_chain, regular_chain, regular_chain]](/support/helpjp/helpview.aspx?si=6432/file06439/math142.png)
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![[[x], [x-1, y], [x-2, y-1, z]]](/support/helpjp/helpview.aspx?si=6432/file06439/math149.png)
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![[2, 1, 0]](/support/helpjp/helpview.aspx?si=6432/file06439/math156.png)
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These regular chains are a surface, a line, and a point respectively.
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