 RegularChains[ChainTools] - Maple Programming Help

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RegularChains[ChainTools]

 Dimension
 dimension of a regular chain

 Calling Sequence Dimension(rc, R)

Parameters

 rc - regular chain of R R - polynomial ring

Description

 • 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.
 • 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](..).

Examples

 > $\mathrm{with}\left(\mathrm{RegularChains}\right):$
 > $\mathrm{with}\left(\mathrm{ChainTools}\right):$
 > $R≔\mathrm{PolynomialRing}\left(\left[x,y,a,b,c,d,g,h\right]\right)$
 ${R}{≔}{\mathrm{polynomial_ring}}$ (1)
 > $\mathrm{sys}≔\left\{ax+by-g,cx+dy-h\right\}$
 ${\mathrm{sys}}{≔}\left\{{a}{}{x}{+}{b}{}{y}{-}{g}{,}{c}{}{x}{+}{d}{}{y}{-}{h}\right\}$ (2)
 > $\mathrm{decl}≔\mathrm{Triangularize}\left(\mathrm{sys},R,'\mathrm{output}'='\mathrm{lazard}'\right)$
 ${\mathrm{decl}}{≔}\left[{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}\right]$ (3)
 > $\mathrm{map}\left(\mathrm{Equations},\mathrm{decl},R\right)$
 $\left[\left[{c}{}{x}{+}{d}{}{y}{-}{h}{,}\left({d}{}{a}{-}{b}{}{c}\right){}{y}{-}{h}{}{a}{+}{c}{}{g}\right]{,}\left[{c}{}{x}{+}{d}{}{y}{-}{h}{,}{d}{}{a}{-}{b}{}{c}{,}{h}{}{b}{-}{d}{}{g}\right]{,}\left[{a}{}{x}{+}{b}{}{y}{-}{g}{,}{d}{}{y}{-}{h}{,}{c}\right]{,}\left[{d}{}{y}{-}{h}{,}{a}{,}{h}{}{b}{-}{d}{}{g}{,}{c}\right]{,}\left[{c}{}{x}{-}{h}{,}{h}{}{a}{-}{c}{}{g}{,}{b}{,}{d}\right]{,}\left[{a}{}{x}{+}{b}{}{y}{-}{g}{,}{c}{,}{d}{,}{h}\right]{,}\left[{c}{}{x}{+}{d}{}{y}{,}{d}{}{a}{-}{b}{}{c}{,}{g}{,}{h}\right]{,}\left[{b}{}{y}{-}{g}{,}{a}{,}{c}{,}{d}{,}{h}\right]{,}\left[{y}{,}{a}{,}{c}{,}{g}{,}{h}\right]{,}\left[{x}{,}{b}{,}{d}{,}{g}{,}{h}\right]{,}\left[{a}{,}{b}{,}{c}{,}{d}{,}{g}{,}{h}\right]\right]$ (4)

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.

 > $\mathrm{map}\left(\mathrm{Dimension},\mathrm{decl},R\right)$
 $\left[{6}{,}{5}{,}{5}{,}{4}{,}{4}{,}{4}{,}{4}{,}{3}{,}{3}{,}{3}{,}{2}\right]$ (5)

Here is another simple example with a triangular decomposition containing regular chains of different dimensions.

 > $R≔\mathrm{PolynomialRing}\left(\left[x,y,z\right]\right)$
 ${R}{≔}{\mathrm{polynomial_ring}}$ (6)
 > $\mathrm{sys}≔\left\{x\left(\left(x-1\right)\left(y-1\right)+\left(x-2\right)y\right),x\left(x-1\right)z,x\left(x-1\right)\left(x-2\right)\right\}$
 ${\mathrm{sys}}{≔}\left\{{x}{}\left(\left({x}{-}{1}\right){}\left({y}{-}{1}\right){+}\left({x}{-}{2}\right){}{y}\right){,}{x}{}\left({x}{-}{1}\right){}{z}{,}{x}{}\left({x}{-}{1}\right){}\left({x}{-}{2}\right)\right\}$ (7)
 > $\mathrm{dec}≔\mathrm{Triangularize}\left(\mathrm{sys},R\right)$
 ${\mathrm{dec}}{≔}\left[{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}\right]$ (8)
 > $\mathrm{map}\left(\mathrm{Equations},\mathrm{dec},R\right)$
 $\left[\left[{x}\right]{,}\left[{x}{-}{1}{,}{y}\right]{,}\left[{x}{-}{2}{,}{y}{-}{1}{,}{z}\right]\right]$ (9)
 > $\mathrm{map}\left(\mathrm{Dimension},\mathrm{dec},R\right)$
 $\left[{2}{,}{1}{,}{0}\right]$ (10)

These regular chains are a surface, a line, and a point respectively.