PrimaryDecomposition - Maple Help

PolynomialIdeals

 PrimaryDecomposition
 compute a primary decomposition of an ideal
 PrimeDecomposition
 compute a prime decomposition of the radical of an ideal

 Calling Sequence PrimaryDecomposition(J, k, removeredundant=b) PrimeDecomposition(J, k, removeredundant=b)

Parameters

 J - polynomial ideal k - (optional) field extension

Options

 removeredundant=b, where b is true or false
 If removeredundant=true, or just removeredundant is specified, then the resulting decomposition is irredundant, i.e., no component can be omitted without changing the intersection. This is not the case by default, for efficiency reasons.

Description

 • The PrimaryDecomposition command constructs a finite sequence of primary ideals whose intersection equals the input J.  Likewise the PrimeDecomposition command constructs a sequence of prime ideals whose intersection is equal to the radical of J. Calling PrimeDecomposition(J) is faster but otherwise equivalent to calling PrimaryDecomposition(Radical(J)).
 • By default, ideals are factored over the domain implied by their coefficients - usually the rationals or the integers mod p.  Additional field extensions can be specified with an optional second argument k, which can be a single RootOf or radical, or a list or set of RootOfs and radicals.
 • The output of these commands is not canonical, and may not be unique. However, a Groebner basis is stored for each ideal in the sequence so the Simplify command can be used at no additional cost.
 • The algorithms employed by these commands require polynomials over a perfect field.  Infinite fields of positive characteristic are not supported. Over finite fields, only zero-dimensional ideals can be handled because the dimension reducing process generates infinite fields.

Examples

 > $\mathrm{with}\left(\mathrm{PolynomialIdeals}\right):$
 > $J≔⟨{x}^{2},xy+x⟩$
 ${J}{≔}⟨{{x}}^{{2}}{,}{x}{}{y}{+}{x}⟩$ (1)
 > $\mathrm{PrimeDecomposition}\left(J\right)$
 $⟨{x}⟩$ (2)
 > $\mathrm{PrimaryDecomposition}\left(J\right)$
 $⟨{x}⟩{,}⟨{{x}}^{{2}}{,}{y}{+}{1}⟩$ (3)
 > $J≔⟨{x}^{2}-2,{y}^{2}+1,{z}^{2}+2⟩$
 ${J}{≔}⟨{{x}}^{{2}}{-}{2}{,}{{y}}^{{2}}{+}{1}{,}{{z}}^{{2}}{+}{2}⟩$ (4)
 > $P≔\mathrm{PrimaryDecomposition}\left(J\right)$
 ${P}{≔}⟨{{x}}^{{2}}{-}{2}{,}{{y}}^{{2}}{+}{1}{,}{{z}}^{{2}}{+}{2}{,}{-}{x}{}{y}{+}{z}⟩{,}⟨{{x}}^{{2}}{-}{2}{,}{{y}}^{{2}}{+}{1}{,}{{z}}^{{2}}{+}{2}{,}{x}{}{y}{+}{z}⟩$ (5)
 > $\mathrm{Intersect}\left(P\right)$
 $⟨{{x}}^{{2}}{-}{2}{,}{{y}}^{{2}}{+}{1}{,}{{z}}^{{2}}{+}{2}⟩$ (6)
 > $K≔⟨J,\mathrm{characteristic}=5⟩$
 ${K}{≔}⟨{{x}}^{{2}}{+}{3}{,}{{y}}^{{2}}{+}{1}{,}{{z}}^{{2}}{+}{2}⟩$ (7)
 > $P≔\mathrm{PrimaryDecomposition}\left(K\right)$
 ${P}{≔}⟨{y}{+}{2}{,}{z}{+}{2}{}{x}{,}{{x}}^{{2}}{+}{3}{,}{{y}}^{{2}}{+}{1}{,}{{z}}^{{2}}{+}{2}⟩{,}⟨{y}{+}{2}{,}{z}{+}{3}{}{x}{,}{{x}}^{{2}}{+}{3}{,}{{y}}^{{2}}{+}{1}{,}{{z}}^{{2}}{+}{2}⟩{,}⟨{y}{+}{3}{,}{z}{+}{2}{}{x}{,}{{x}}^{{2}}{+}{3}{,}{{y}}^{{2}}{+}{1}{,}{{z}}^{{2}}{+}{2}⟩{,}⟨{y}{+}{3}{,}{z}{+}{3}{}{x}{,}{{x}}^{{2}}{+}{3}{,}{{y}}^{{2}}{+}{1}{,}{{z}}^{{2}}{+}{2}⟩$ (8)
 > $P≔\mathrm{Simplify}\left(P\right)$
 ${P}{≔}⟨{y}{+}{2}{,}{z}{+}{2}{}{x}{,}{{x}}^{{2}}{+}{3}⟩{,}⟨{y}{+}{2}{,}{z}{+}{3}{}{x}{,}{{x}}^{{2}}{+}{3}⟩{,}⟨{y}{+}{3}{,}{z}{+}{2}{}{x}{,}{{x}}^{{2}}{+}{3}⟩{,}⟨{y}{+}{3}{,}{z}{+}{3}{}{x}{,}{{x}}^{{2}}{+}{3}⟩$ (9)
 > $\mathrm{Intersect}\left(P\right)$
 $⟨{{x}}^{{2}}{+}{3}{,}{{y}}^{{2}}{+}{1}{,}{{z}}^{{2}}{+}{2}⟩$ (10)
 > $L≔⟨x{z}^{2}-2xy,{y}^{2}-{z}^{2}x⟩$
 ${L}{≔}⟨{-}{x}{}{{z}}^{{2}}{+}{{y}}^{{2}}{,}{x}{}{{z}}^{{2}}{-}{2}{}{x}{}{y}⟩$ (11)
 > $P≔\mathrm{PrimaryDecomposition}\left(L\right)$
 ${P}{≔}⟨{x}{,}{{y}}^{{2}}⟩{,}⟨{y}{,}{{z}}^{{2}}⟩{,}⟨{-}{{z}}^{{2}}{+}{4}{}{x}{,}{-}{{z}}^{{2}}{+}{2}{}{y}⟩$ (12)
 > $\mathrm{IdealContainment}\left(L,\mathrm{Intersect}\left(P\right),L\right)$
 ${\mathrm{true}}$ (13)
 > $H≔⟨{x}^{2}+{y}^{2}-1,x+y⟩$
 ${H}{≔}⟨{x}{+}{y}{,}{{x}}^{{2}}{+}{{y}}^{{2}}{-}{1}⟩$ (14)
 > $\mathrm{PrimeDecomposition}\left(H\right)$
 $⟨{x}{+}{y}{,}{{x}}^{{2}}{+}{{y}}^{{2}}{-}{1}⟩$ (15)
$k=Q\left(\mathrm{alpha}\right)=Q\left(\sqrt{2}\right)$
 > $\mathrm{alias}\left(\mathrm{\alpha }=\mathrm{RootOf}\left({z}^{2}-2\right)\right)$
 ${\mathrm{\alpha }}$ (16)
 > $P≔\mathrm{PrimeDecomposition}\left(H,\mathrm{\alpha }\right)$
 ${P}{≔}⟨{x}{+}{y}{,}{2}{}{x}{-}{\mathrm{\alpha }}{,}{{x}}^{{2}}{+}{{y}}^{{2}}{-}{1}⟩{,}⟨{x}{+}{y}{,}{2}{}{x}{+}{\mathrm{\alpha }}{,}{{x}}^{{2}}{+}{{y}}^{{2}}{-}{1}⟩$ (17)
 > $\mathrm{Simplify}\left(P\right)$
 $⟨{2}{}{x}{-}{\mathrm{\alpha }}{,}{\mathrm{\alpha }}{+}{2}{}{y}⟩{,}⟨{2}{}{x}{+}{\mathrm{\alpha }}{,}{-}{\mathrm{\alpha }}{+}{2}{}{y}⟩$ (18)

Redundant components are not removed by default:

 > $\mathrm{PrimaryDecomposition}\left(⟨xy,{x}^{2}+x⟩\right)$
 $⟨{x}⟩{,}⟨{x}{,}{y}⟩{,}⟨{y}{,}{x}{+}{1}⟩$ (19)
 > $\mathrm{PrimaryDecomposition}\left(⟨xy,{x}^{2}+x⟩,'\mathrm{removeredundant}'\right)$
 $⟨{x}⟩{,}⟨{y}{,}{x}{+}{1}⟩$ (20)

References

 Gianni, P.; Trager, B.; and Zacharias, G. "Grobner bases and primary decompositions of polynomial ideals." J. Symbolic Comput. Vol. 6, (1988): 149-167.

Compatibility

 • The PolynomialIdeals[PrimaryDecomposition] and PolynomialIdeals[PrimeDecomposition] commands were updated in Maple 2021.
 • The removeredundant option was introduced in Maple 2021.