The Contract command contracts the image of an ideal in ${k\left(U\right)}_{\mathrm{XU}}$ back to the polynomial ring $k_X$.

The typical use of this command is to contract the result of a zero-dimensional decomposition back to the original ring. This is part of a process used to extend algorithms for zero-dimensional ideals to ideals of positive dimension. For additional details, see the help page for ZeroDimensionalDecomposition.

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

$\mathrm{Jxy}≔⟨z^2{x}^2-y\,z^{3}xy\,\mathrm{variables}=\left\{x\,y\right\}⟩$

$\mathrm{Jxy}≔⟨z^{3}xy\,z^2{x}^2-y⟩$

$J≔\mathrm{Contract}\left(\mathrm{Jxy}\,\left\{x\,y\,z\right\}\right)$

$J≔⟨x^3\,y^2\,xy\,z^2{x}^2-y⟩$

$\mathrm{IdealInfo}:-\mathrm{Variables}\left(J\right)$

$\left\{x\,y\,z\right\}$

$K≔\mathrm{Intersect}\left(J\,⟨y^2-xz⟩\right)$

$K≔⟨-xyz+y^3\,-x^{3}z+x^2{y}^2⟩$

$\mathrm{zdd}≔\mathrm{ZeroDimensionalDecomposition}\left(K\right)$

$\mathrm{zdd}≔⟨-xz+y^2⟩,⟨x^2\,-xyz+y^3⟩$

$\mathrm{seq}\left(\mathrm{HilbertDimension}\left(i\right)\,i=\left[\mathrm{zdd}\right]\right)$

$0,0$

$\mathrm{seq}\left(\mathrm{IdealInfo}:-\mathrm{Variables}\left(i\right)\,i=\left[\mathrm{zdd}\right]\right)$

$\left\{y\right\},\left\{x\,y\right\}$

$C≔\mathrm{seq}\left(\mathrm{Contract}\left(i\,\left\{x\,y\,z\right\}\right)\,i=\left[\mathrm{zdd}\right]\right)$

$C≔⟨-xz+y^2⟩,⟨x^2\,-xyz+y^3⟩$

$\mathrm{seq}\left(\mathrm{IdealInfo}:-\mathrm{Variables}\left(i\right)\,i=\left[C\right]\right)$

$\left\{x\,y\,z\right\},\left\{x\,y\,z\right\}$

$L≔\mathrm{Intersect}\left(C\right)$

$L≔⟨xyz-y^3\,x^{3}z-x^2{y}^2⟩$

$\mathrm{IdealContainment}\left(K\,L\,K\right)$

$\mathrm{true}$

Becker, T., and Weispfenning, V. Groebner Bases. New York: Springer-Verlag, 1993.