EquidimensionalDecomposition - Maple Help

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PolynomialIdeals

 EquidimensionalDecomposition
 Decompose an ideal into ideals of distinct dimension

 Calling Sequence EquidimensionalDecomposition(J)

Parameters

 J - a polynomial ideal

Description

 • The EquidimensionalDecomposition command computes a sequence of ideals of distinct Hilbert dimension whose intersection is equal to the original ideal. Assuming there are no embedded primes, the prime components of each ideal in the sequence have the same dimension also. In general this decomposition is not unique.
 • This function is part of the PolynomialIdeals package, and can be used in the form EquidimensionalDecomposition(..) only after executing the command with(PolynomialIdeals).  However, it can always be accessed through the long form of the command using PolynomialIdeals[EquidimensionalDecomposition](..).

Examples

In the example below, the variety is composed of five points (dimension zero), three curves (dimension one), and one surface (dimension two). The equidimensional decomposition places all of the points, all of the curves, and all of the surfaces into separate ideals.

 > $\mathrm{with}\left(\mathrm{PolynomialIdeals}\right):$
 > $J≔⟨{z}^{4}{x}^{3}+{z}^{4}{y}^{2}-{z}^{5}-{z}^{4}x-z{x}^{4}-x{y}^{2}z+{z}^{2}x+z{x}^{2},{x}^{2}{y}^{2}-z{x}^{2}+zx-{y}^{2}z-{x}^{3}z-yz+{x}^{3}y-xy+{x}^{5}-{x}^{3}+{y}^{3}+{z}^{2},{z}^{3}{x}^{4}+{z}^{3}x{y}^{2}-{z}^{3}{x}^{3}-{x}^{5}-{y}^{2}{z}^{3}-{z}^{3}{x}^{2}-{z}^{4}x-{x}^{2}{y}^{2}+{x}^{4}+{z}^{3}x+{z}^{4}+x{y}^{2}+{x}^{3}+z{x}^{2}-{x}^{2}-zx⟩:$
 > $E≔\mathrm{EquidimensionalDecomposition}\left(J\right)$
 ${E}{≔}⟨{{z}}^{{2}}{,}{{y}}^{{2}}{}{z}{+}{y}{}{z}{,}{{y}}^{{6}}{+}{2}{}{{y}}^{{5}}{+}{{y}}^{{4}}{,}{{x}}^{{3}}{}{z}{}{y}{-}{x}{}{y}{}{z}{+}{y}{}{z}{,}{{x}}^{{4}}{+}{{x}}^{{3}}{}{y}{-}{{x}}^{{3}}{}{z}{+}{x}{}{{y}}^{{2}}{+}{{y}}^{{3}}{-}{{x}}^{{2}}{-}{x}{}{y}{,}{{x}}^{{3}}{}{{y}}^{{2}}{+}{{x}}^{{3}}{}{y}{-}{{x}}^{{3}}{}{z}{+}{{y}}^{{4}}{-}{x}{}{{y}}^{{2}}{+}{{y}}^{{3}}{-}{x}{}{y}{+}{z}{}{x}{+}{y}{}{z}⟩{,}⟨{y}{}{{z}}^{{6}}{+}{{z}}^{{6}}{+}{{y}}^{{2}}{-}{y}{}{z}{+}{y}{-}{z}{,}{{x}}^{{3}}{}{{z}}^{{6}}{-}{x}{}{{z}}^{{6}}{-}{{z}}^{{7}}{+}{{z}}^{{6}}{+}{{x}}^{{3}}{}{y}{-}{{x}}^{{3}}{}{z}{-}{x}{}{y}{+}{z}{}{x}{-}{y}{}{z}{+}{{z}}^{{2}}{+}{y}{-}{z}{,}{y}{}{{z}}^{{9}}{+}{{z}}^{{9}}{-}{x}{}{y}{}{{z}}^{{6}}{-}{x}{}{{z}}^{{6}}{-}{{x}}^{{3}}{}{{z}}^{{3}}{-}{{z}}^{{4}}{}{y}{+}{{x}}^{{4}}{+}{{z}}^{{3}}{}{x}{+}{y}{}{{z}}^{{3}}{+}{x}{}{y}{}{z}{-}{{x}}^{{2}}{-}{x}{}{y}⟩{,}⟨{{x}}^{{3}}{+}{{y}}^{{2}}{-}{x}{-}{z}⟩$ (1)
 > $\mathrm{seq}\left(\mathrm{HilbertDimension}\left(i\right),i=E\right)$
 ${0}{,}{1}{,}{2}$ (2)
 > $\mathrm{seq}\left(\mathrm{map}\left(\mathrm{HilbertDimension},\left[\mathrm{PrimaryDecomposition}\left(i\right)\right],\left\{x,y,z\right\}\right),i=E\right)$
 $\left[{0}{,}{0}{,}{0}{,}{0}{,}{0}\right]{,}\left[{1}{,}{1}{,}{1}\right]{,}\left[{2}\right]$ (3)

The next example illustrates what happens when embedded primes are present.

 > $K≔⟨{x}^{2}-y,{x}^{3}-yzw⟩$
 ${K}{≔}⟨{{x}}^{{2}}{-}{y}{,}{-}{y}{}{z}{}{w}{+}{{x}}^{{3}}⟩$ (4)
 > $E≔\mathrm{EquidimensionalDecomposition}\left(K\right)$
 ${E}{≔}⟨{w}{,}{{y}}^{{2}}{,}{x}{}{y}{,}{{x}}^{{2}}{-}{y}⟩{,}⟨{{x}}^{{2}}{-}{y}{,}{y}{}{z}{}{w}{-}{x}{}{y}⟩$ (5)
 > $\mathrm{seq}\left(\mathrm{HilbertDimension}\left(i,\left\{w,x,y,z\right\}\right),i=E\right)$
 ${1}{,}{2}$ (6)
 > $\mathrm{seq}\left(\mathrm{map}\left(\mathrm{HilbertDimension},\left[\mathrm{PrimaryDecomposition}\left(i\right)\right],\left\{w,x,y,z\right\}\right),i=E\right)$
 $\left[{1}\right]{,}\left[{2}{,}{2}{,}{1}\right]$ (7)

References

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

Compatibility

 • The PolynomialIdeals[EquidimensionalDecomposition] command was updated in Maple 16.