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](..).

$\mathrm{with}\left(\mathrm{RegularChains}\right)\:$

$\mathrm{with}\left(\mathrm{ChainTools}\right)\:$

$R≔\mathrm{PolynomialRing}\left(\left[x\,a\right]\,\left\{b\,c\right\}\right)$

$R≔\mathrm{polynomial\_ring}$

$\mathrm{sys}≔\left[a{x}^2+bx+c\right]$

$\mathrm{sys}≔\left[a{x}^2+bx+c\right]$

$\mathrm{dec}≔\mathrm{Triangularize}\left(\mathrm{sys}\,R\,\mathrm{output}=\mathrm{lazard}\right)$

$\mathrm{dec}≔\left[\mathrm{regular\_chain}\,\mathrm{regular\_chain}\right]$

$\mathrm{map}\left(\mathrm{Equations}\,\mathrm{dec}\,R\right)$

$\left[\left[a{x}^2+bx+c\right]\,\left[bx+c\,a\right]\right]$

$\mathrm{map}\left(\mathrm{Dimension}\,\mathrm{dec}\,R\right)$

$\left[1\,0\right]$

$\mathrm{map}\left(\mathrm{NumberOfSolutions}\,\mathrm{dec}\,R\right)$

$\left[\mathrm{\infty }\,1\right]$