compute a primary decomposition of an ideal

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PolynomialIdeals[PrimaryDecomposition] - compute a primary decomposition of an ideal

PolynomialIdeals[PrimeDecomposition] - compute a prime decomposition of the radical of an ideal

	Calling Sequence
	PrimaryDecomposition(J, k) PrimeDecomposition(J, k)

Parameters

J	-	polynomial ideal
k	-	(optional) field extension

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.

Compatibility

•	The PolynomialIdeals[PrimaryDecomposition] and PolynomialIdeals[PrimeDecomposition] commands were updated in Maple 16.

Examples

$POLYNOMIALIDEAL(x^2, x*y+x, variables = {x, y}, characteristic = 0, known_groebner_bases = (table( [ ] )))$

(1)

(2)

$POLYNOMIALIDEAL(x, variables = {x, y}, characteristic = 0, known_groebner_bases = (table( [( plex(x) ) = [[1, x, x]], ( lexdeg([x], [y]) ) = [[1, x, x]], ( tdeg(y, x) ) = [[1, x, x]], ( plex(y, x) ) = [[1, x, x]] ] ))), POLYNOMIALIDEAL(x^2, y+1, variables = {x, y}, characteristic = 0, known_groebner_bases = (table( [( tdeg(y, x) ) = [[1, y, y+1], [1, x^2, x^2]], ( plex(y, x) ) = [[1, x^2, x^2], [1, y, y+1]] ] )))$

(3)

$POLYNOMIALIDEAL(x^2-2, y^2+1, z^2+2, variables = {x, y, z}, characteristic = 0, known_groebner_bases = (table( [ ] )))$

(4)

$POLYNOMIALIDEAL(z-x*y, x^2-2, y^2+1, z^2+2, variables = {x, y, z}, characteristic = 0, known_groebner_bases = (table( [( plex(z, y, x) ) = [[1, x^2, x^2-2], [1, y^2, y^2+1], [1, z, z-x*y]] ] ))), POLYNOMIALIDEAL(z+x*y, x^2-2, y^2+1, z^2+2, variables = {x, y, z}, characteristic = 0, known_groebner_bases = (table( [( plex(z, y, x) ) = [[1, x^2, x^2-2], [1, y^2, y^2+1], [1, z, z+x*y]] ] )))$

(5)

$POLYNOMIALIDEAL(x^2-2, y^2+1, z^2+2, variables = {x, y, z}, characteristic = 0, known_groebner_bases = (table( [( plex(z, y, x) ) = [[1, x^2, x^2-2], [1, y^2, y^2+1], [1, z^2, z^2+2]], ( tdeg(z, y, x) ) = [[1, x^2, x^2-2], [1, y^2, y^2+1], [1, z^2, z^2+2]] ] )))$

(6)

$POLYNOMIALIDEAL(x^2+3, y^2+1, z^2+2, variables = {x, y, z}, characteristic = 5, known_groebner_bases = (table( [ ] )))$

(7)

(8)

(9)

$POLYNOMIALIDEAL(x^2+3, y^2+1, z^2+2, variables = {x, y, z}, characteristic = 5, known_groebner_bases = (table( [( plex(z, y, x) ) = [[1, x^2, x^2+3], [1, y^2, y^2+1], [1, z^2, z^2+2]], ( tdeg(z, y, x) ) = [[1, x^2, x^2+3], [1, y^2, y^2+1], [1, z^2, z^2+2]] ] )))$

(10)

$POLYNOMIALIDEAL(y^2-z^2*x, z^2*x-2*x*y, variables = {x, y, z}, characteristic = 0, known_groebner_bases = (table( [ ] )))$

(11)

(12)

(13)

$POLYNOMIALIDEAL(x+y, x^2+y^2-1, variables = {x, y}, characteristic = 0, known_groebner_bases = (table( [ ] )))$

(14)

$POLYNOMIALIDEAL(x+y, x^2+y^2-1, variables = {x, y}, characteristic = 0, known_groebner_bases = (table( [( tdeg(y, x) ) = [[1, y, x+y], [2, x^2, 2*x^2-1]], ( plex(y, x) ) = [[2, x^2, 2*x^2-1], [1, y, x+y]] ] )))$

(15)

(16)

$POLYNOMIALIDEAL(x+y, 2*x-alpha, x^2+y^2-1, variables = {x, y}, characteristic = 0, known_groebner_bases = (table( [( plex(y, x) ) = [[2, x, 2*x-RootOf(_Z^2-2)], [2, y, RootOf(_Z^2-2)+2*y]] ] ))), POLYNOMIALIDEAL(x+y, 2*x+alpha, x^2+y^2-1, variables = {x, y}, characteristic = 0, known_groebner_bases = (table( [( plex(y, x) ) = [[2, x, 2*x+RootOf(_Z^2-2)], [2, y, -RootOf(_Z^2-2)+2*y]] ] )))$

(17)

$POLYNOMIALIDEAL(2*x-alpha, alpha+2*y, variables = {x, y}, characteristic = 0, known_groebner_bases = (table( [( plex(y, x) ) = [[2, x, 2*x-RootOf(_Z^2-2)], [2, y, RootOf(_Z^2-2)+2*y]] ] ))), POLYNOMIALIDEAL(2*x+alpha, -alpha+2*y, variables = {x, y}, characteristic = 0, known_groebner_bases = (table( [( plex(y, x) ) = [[2, x, 2*x+RootOf(_Z^2-2)], [2, y, -RootOf(_Z^2-2)+2*y]] ] )))$

(18)

	See Also
	Groebner[Solve], PolynomialIdeals, PolynomialIdeals[IdealContainment], PolynomialIdeals[Intersect], PolynomialIdeals[IsPrimary], PolynomialIdeals[IsPrime], PolynomialIdeals[Radical], PolynomialIdeals[Simplify], PolynomialIdeals[ZeroDimensionalDecomposition]

References

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

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