RegularChains[ChainTools] - Maple Programming Help

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RegularChains[ChainTools]

 NumberOfSolutions
 number of solutions of a regular chain

 Calling Sequence NumberOfSolutions(rc, R)

Parameters

 rc - regular chain of R R - polynomial ring

Description

 • The command NumberOfSolutions(rc, R) returns the number of complex solutions of rc.
 • If rc has a positive dimension, then infinity is returned.
 • If rc has dimension zero, the number of roots is returned.
 • This command is part of the RegularChains[ChainTools] package, so it can be used in the form NumberOfSolutions(..) 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][NumberOfSolutions](..).

Examples

 > $\mathrm{with}\left(\mathrm{RegularChains}\right):$
 > $\mathrm{with}\left(\mathrm{ChainTools}\right):$
 > $R≔\mathrm{PolynomialRing}\left(\left[x,a\right],\left\{b,c\right\}\right)$
 ${R}{:=}{\mathrm{polynomial_ring}}$ (1)
 > $\mathrm{sys}≔\left[a{x}^{2}+bx+c\right]$
 ${\mathrm{sys}}{:=}\left[{a}{}{{x}}^{{2}}{+}{b}{}{x}{+}{c}\right]$ (2)
 > $\mathrm{dec}≔\mathrm{Triangularize}\left(\mathrm{sys},R,\mathrm{output}=\mathrm{lazard}\right)$
 ${\mathrm{dec}}{:=}\left[{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}\right]$ (3)
 > $\mathrm{map}\left(\mathrm{Equations},\mathrm{dec},R\right)$
 $\left[\left[{a}{}{{x}}^{{2}}{+}{b}{}{x}{+}{c}\right]{,}\left[{b}{}{x}{+}{c}{,}{a}\right]\right]$ (4)
 > $\mathrm{map}\left(\mathrm{Dimension},\mathrm{dec},R\right)$
 $\left[{1}{,}{0}\right]$ (5)
 > $\mathrm{map}\left(\mathrm{NumberOfSolutions},\mathrm{dec},R\right)$
 $\left[{\mathrm{∞}}{,}{1}\right]$ (6)