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](..).

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

$J≔⟨z^4{x}^3+z^4{y}^2-z^5-z^{4}x-z{x}^4-x{y}^{2}z+z^{2}x+z{x}^2\,x^2{y}^2-z{x}^2+zx-y^{2}z-x^{3}z-yz+x^{3}y-xy+x^5-x^3+y^3+z^2\,z^3{x}^4+z^{3}x{y}^2-z^3{x}^3-x^5-y^2{z}^3-z^3{x}^2-z^{4}x-x^2{y}^2+x^4+z^{3}x+z^4+x{y}^2+x^3+z{x}^2-x^2-zx⟩\:$

$E≔\mathrm{EquidimensionalDecomposition}\left(J\right)$

$E≔⟨z^2\,y^{2}z+yz\,y^6+2y^5+y^4\,x^{3}zy-xyz+yz\,x^4+x^{3}y-x^{3}z+x{y}^2+y^3-x^2-xy\,x^3{y}^2+x^{3}y-x^{3}z+y^4-x{y}^2+y^3-xy+xz+yz⟩,⟨y{z}^6+z^6+y^2-yz+y-z\,x^3{z}^6-x{z}^6-z^7+z^6+x^{3}y-x^{3}z-xy+xz-yz+z^2+y-z\,y{z}^9+z^9-xy{z}^6-x{z}^6-x^3{z}^3-z^{4}y+x^4+x{z}^3+y{z}^3+xyz-x^2-xy⟩,⟨x^3+y^2-x-z⟩$

$\mathrm{seq}\left(\mathrm{HilbertDimension}\left(i\right)\,i=E\right)$

$0,1,2$

$\mathrm{seq}\left(\mathrm{map}\left(\mathrm{HilbertDimension}\,\left[\mathrm{PrimaryDecomposition}\left(i\right)\right]\,\left\{x\,y\,z\right\}\right)\,i=E\right)$

$\left[0\,0\,0\,0\,0\right],\left[1\,1\,1\right],\left[2\right]$

$K≔⟨x^2-y\,x^3-yzw⟩$

$K≔⟨x^2-y\,-yzw+x^3⟩$

$E≔\mathrm{EquidimensionalDecomposition}\left(K\right)$

$E≔⟨w\,y^2\,xy\,x^2-y⟩,⟨x^2-y\,yzw-xy⟩$

$\mathrm{seq}\left(\mathrm{HilbertDimension}\left(i\,\left\{w\,x\,y\,z\right\}\right)\,i=E\right)$

$1,2$

$\mathrm{seq}\left(\mathrm{map}\left(\mathrm{HilbertDimension}\,\left[\mathrm{PrimaryDecomposition}\left(i\right)\right]\,\left\{w\,x\,y\,z\right\}\right)\,i=E\right)$

$\left[1\right],\left[2\,2\,1\right]$

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

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