Contract - Maple Help

PolynomialIdeals

 Contract
 contract an ideal to a larger ring

 Calling Sequence Contract(J, X)

Parameters

 J - polynomial ideal X - set of variable names

Description

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{PolynomialIdeals}\right):$
 > $\mathrm{Jxy}≔⟨{z}^{2}{x}^{2}-y,{z}^{3}xy,\mathrm{variables}=\left\{x,y\right\}⟩$
 ${\mathrm{Jxy}}{≔}⟨{{z}}^{{3}}{}{x}{}{y}{,}{{z}}^{{2}}{}{{x}}^{{2}}{-}{y}⟩$ (1)
 > $J≔\mathrm{Contract}\left(\mathrm{Jxy},\left\{x,y,z\right\}\right)$
 ${J}{≔}⟨{{x}}^{{3}}{,}{{y}}^{{2}}{,}{x}{}{y}{,}{{z}}^{{2}}{}{{x}}^{{2}}{-}{y}⟩$ (2)
 > $\mathrm{IdealInfo}:-\mathrm{Variables}\left(J\right)$
 $\left\{{x}{,}{y}{,}{z}\right\}$ (3)
 > $K≔\mathrm{Intersect}\left(J,⟨{y}^{2}-xz⟩\right)$
 ${K}{≔}⟨{-}{x}{}{y}{}{z}{+}{{y}}^{{3}}{,}{-}{{x}}^{{3}}{}{z}{+}{{x}}^{{2}}{}{{y}}^{{2}}⟩$ (4)
 > $\mathrm{zdd}≔\mathrm{ZeroDimensionalDecomposition}\left(K\right)$
 ${\mathrm{zdd}}{≔}⟨{-}{x}{}{z}{+}{{y}}^{{2}}⟩{,}⟨{{x}}^{{2}}{,}{-}{x}{}{y}{}{z}{+}{{y}}^{{3}}⟩$ (5)
 > $\mathrm{seq}\left(\mathrm{HilbertDimension}\left(i\right),i=\left[\mathrm{zdd}\right]\right)$
 ${0}{,}{0}$ (6)
 > $\mathrm{seq}\left(\mathrm{IdealInfo}:-\mathrm{Variables}\left(i\right),i=\left[\mathrm{zdd}\right]\right)$
 $\left\{{y}\right\}{,}\left\{{x}{,}{y}\right\}$ (7)
 > $C≔\mathrm{seq}\left(\mathrm{Contract}\left(i,\left\{x,y,z\right\}\right),i=\left[\mathrm{zdd}\right]\right)$
 ${C}{≔}⟨{-}{x}{}{z}{+}{{y}}^{{2}}⟩{,}⟨{{x}}^{{2}}{,}{-}{x}{}{y}{}{z}{+}{{y}}^{{3}}⟩$ (8)
 > $\mathrm{seq}\left(\mathrm{IdealInfo}:-\mathrm{Variables}\left(i\right),i=\left[C\right]\right)$
 $\left\{{x}{,}{y}{,}{z}\right\}{,}\left\{{x}{,}{y}{,}{z}\right\}$ (9)
 > $L≔\mathrm{Intersect}\left(C\right)$
 ${L}{≔}⟨{x}{}{y}{}{z}{-}{{y}}^{{3}}{,}{{x}}^{{3}}{}{z}{-}{{x}}^{{2}}{}{{y}}^{{2}}⟩$ (10)
 > $\mathrm{IdealContainment}\left(K,L,K\right)$
 ${\mathrm{true}}$ (11)

References

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