The command IsIncluded(rc1, rc2, R) returns true if the saturated ideal of rc1 is detected to be contained in that of rc2, false otherwise, where rc1 and rc2 are regular chains of R.

The answer is true if the following conditions hold.

(1) all equations of rc1 are reduced to zero by rc2

(2) the initials of rc1 are regular modulo rc2

The answer is also true if the following conditions hold.

(1) all equations of rc1 are reduced to zero by rc2

(2) the regular chain rc1 is primitive, that is, it generates its saturated ideal.

Other criteria are implemented.  Some inclusions are not detected by any of those criteria. When they all fail, then false is returned.

Even though there exists a general algorithm for deciding whether the saturated ideal rc1 is contained in that of rc2, this algorithm is not implemented since it too costly to execute in most cases. On the criteria the implemented criteria are in general much less costly to execute.

This command is part of the RegularChains[ChainTools] package, so it can be used in the form IsIncluded(..) 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][IsIncluded](..).

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

$R≔\mathrm{PolynomialRing}\left(\left[x\,y\,z\right]\right)$

$R≔\mathrm{polynomial\_ring}$

$\mathrm{sys}≔\left[x^2+y+z-1\,x+y^2+z-1\,x+y+z^2-1\right]$

$\mathrm{sys}≔\left[x^2+y+z-1\,y^2+x+z-1\,z^2+x+y-1\right]$

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

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

$\mathrm{epdec}≔\mathrm{EquiprojectableDecomposition}\left(\mathrm{dec}\,R\right)$

$\mathrm{epdec}≔\left\{\mathrm{regular\_chain}\,\mathrm{regular\_chain}\right\}$

$$\begin{gathered} \mathbf{for}i\mathbf{to}\mathrm{nops}\left(\mathrm{dec}\right)\mathbf{do} \\ \mathbf{for}j\mathbf{to}\mathrm{nops}\left(\mathrm{epdec}\right)\mathbf{do} \\ T≔\mathrm{dec}\left[i\right]; \\ U≔\mathrm{epdec}\left[j\right]; \\ \mathrm{print}\left(\mathrm{Equations}\left(T\,R\right)\right); \\ \mathrm{print}\left(\mathrm{Equations}\left(U\,R\right)\right); \\ \mathrm{print}\left(\mathrm{IsIncluded}\left(T\,U\,R\right)\right); \\ \mathrm{print}\left(\mathrm{Equations}\left(U\,R\right)\right); \\ \mathrm{print}\left(\mathrm{Equations}\left(T\,R\right)\right); \\ \mathrm{print}\left(\mathrm{IsIncluded}\left(U\,T\,R\right)\right) \\ \mathbf{end}\mathbf{do} \\ \mathbf{end}\mathbf{do} \end{gathered}$$

$\left[x-z\,y-z\,z^2+2z-1\right]$

$\left[2x+z^2-1\,2y+z^2-1\,z^3+z^2-3z+1\right]$

$\mathrm{true}$

$\left[2x+z^2-1\,2y+z^2-1\,z^3+z^2-3z+1\right]$

$\left[x-z\,y-z\,z^2+2z-1\right]$

$\mathrm{false}$

$\left[x-z\,y-z\,z^2+2z-1\right]$

$\left[x+y-1\,y^2-y\,z\right]$

$\mathrm{false}$

$\left[x+y-1\,y^2-y\,z\right]$

$\left[x-z\,y-z\,z^2+2z-1\right]$

$\mathrm{false}$

$\left[x\,y\,z-1\right]$

$\left[2x+z^2-1\,2y+z^2-1\,z^3+z^2-3z+1\right]$

$\mathrm{true}$

$\left[2x+z^2-1\,2y+z^2-1\,z^3+z^2-3z+1\right]$

$\left[x\,y\,z-1\right]$

$\mathrm{false}$

$\left[x\,y\,z-1\right]$

$\left[x+y-1\,y^2-y\,z\right]$

$\mathrm{false}$

$\left[x+y-1\,y^2-y\,z\right]$

$\left[x\,y\,z-1\right]$

$\mathrm{false}$

$\left[x\,y-1\,z\right]$

$\left[2x+z^2-1\,2y+z^2-1\,z^3+z^2-3z+1\right]$

$\mathrm{false}$

$\left[2x+z^2-1\,2y+z^2-1\,z^3+z^2-3z+1\right]$

$\left[x\,y-1\,z\right]$

$\mathrm{false}$

$\left[x\,y-1\,z\right]$

$\left[x+y-1\,y^2-y\,z\right]$

$\mathrm{true}$

$\left[x+y-1\,y^2-y\,z\right]$

$\left[x\,y-1\,z\right]$

$\mathrm{false}$

$\left[x-1\,y\,z\right]$

$\left[2x+z^2-1\,2y+z^2-1\,z^3+z^2-3z+1\right]$

$\mathrm{false}$

$\left[2x+z^2-1\,2y+z^2-1\,z^3+z^2-3z+1\right]$

$\left[x-1\,y\,z\right]$

$\mathrm{false}$

$\left[x-1\,y\,z\right]$

$\left[x+y-1\,y^2-y\,z\right]$

$\mathrm{true}$

$\left[x+y-1\,y^2-y\,z\right]$

$\left[x-1\,y\,z\right]$

$\mathrm{false}$

Xie, Y. "Fast Algorithms, Modular Methods, Parallel Approaches and Software Engineering for Solving Polynomial Systems Symbolically" PhD Thesis, University of Western Ontario, Canada, 2007.