The ZeroDimensionalDecomposition command computes a sequence of zero-dimensional ideals, some of which may lie in extended polynomial rings.  That is, to make the resulting ideals zero-dimensional, ring variables can be moved into the coefficient field.  If the ideals in the resulting sequence are contracted back to the original ring and intersected, you get the original ideal.  In general, this decomposition is not unique.

This command allows you to run algorithms for zero-dimensional ideals on ideals of positive Hilbert dimension.  Be aware that some algorithms do not interact well with the extension and contraction process.  In particular, you cannot use this process to directly test whether an ideal is radical because the decomposition of a radical ideal may contain non-radical components that vanish under contraction and intersection.  Valid applications include solving, testing whether an ideal is prime or primary, and computing prime or primary decompositions or radical of an ideal.

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

$J≔⟨xy-y⟩$

$J≔⟨xy-y⟩$

$\mathrm{zdd}≔\left[\mathrm{ZeroDimensionalDecomposition}\left(J\right)\right]$

$\mathrm{zdd}≔\left[⟨y⟩\,⟨x-1⟩\right]$

$\mathrm{Intersect}\left(\mathrm{op}\left(\mathrm{map}\left(\mathrm{Contract}\,\mathrm{zdd}\,\left\{x\,y\right\}\right)\right)\right)$

$⟨xy-y⟩$

$K≔⟨-2y^3+3x^{3}z\,-y^2{z}^2⟩$

$K≔⟨-y^2{z}^2\,3x^{3}z-2y^3⟩$

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

$\mathrm{zdd}≔\left[⟨x^3\,y^2⟩\,⟨y^8\,z^3\,y^2{z}^2\,z{y}^5\,3x^{3}z-2y^3⟩\right]$

$\mathrm{map}\left(\mathrm{Simplify}@\mathrm{Radical}\,\mathrm{zdd}\right)$

$\left[⟨x\,y⟩\,⟨y\,z⟩\right]$

$\mathrm{Intersect}\left(\mathrm{op}\left(\mathrm{map}\left(\mathrm{Contract}\,\,\left\{x\,y\,z\right\}\right)\right)\right)$

$⟨y\,zx⟩$

$\mathrm{Radical}\left(K\right)$

$⟨y\,zx⟩$

$L≔⟨x-y\,x^3-yzw⟩$

$L≔⟨x-y\,-yzw+x^3⟩$

$\mathrm{ZeroDimensionalDecomposition}\left(L\right)$

$⟨x\,y⟩,⟨y-x\,wz-x^2⟩,⟨w\,x^3\,y-x⟩$

$\mathrm{map}\left(\mathrm{IsRadical}\,\left[\right]\right)$

$\left[\mathrm{true}\,\mathrm{true}\,\mathrm{false}\right]$

$\mathrm{IsRadical}\left(L\right)$

$\mathrm{true}$

$\mathrm{L2}≔⟨x^2-y\,x^3-yzw⟩$

$\mathrm{L2}≔⟨x^2-y\,-yzw+x^3⟩$

$\mathrm{ZeroDimensionalDecomposition}\left(\mathrm{L2}\right)$

$⟨y\,x^2⟩,⟨x^2-y\,wz-x⟩,⟨w\,y^2\,xy\,x^2-y⟩$

$\mathrm{map}\left(\mathrm{IsRadical}\,\left[\right]\right)$

$\left[\mathrm{false}\,\mathrm{true}\,\mathrm{false}\right]$

$\mathrm{IsRadical}\left(\mathrm{L2}\right)$

$\mathrm{false}$

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