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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
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Calling Sequence
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PrimaryDecomposition(J, k)
PrimeDecomposition(J, k)
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Parameters
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J
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polynomial ideal
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k
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(optional) field extension
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Description
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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)).
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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.
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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.
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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.
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Compatibility
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The PolynomialIdeals[PrimaryDecomposition] and PolynomialIdeals[PrimeDecomposition] commands were updated in Maple 16.
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Examples
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![POLYNOMIALIDEAL(x^2, x*y+x, variables = {x, y}, characteristic = 0, known_groebner_bases = (table( [ ] )))](/support/helpjp/helpview.aspx?si=8505/file02595/math122.png)
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![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]] ] )))](/support/helpjp/helpview.aspx?si=8505/file02595/math129.png)
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![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]] ] )))](/support/helpjp/helpview.aspx?si=8505/file02595/math136.png)
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![POLYNOMIALIDEAL(x^2-2, y^2+1, z^2+2, variables = {x, y, z}, characteristic = 0, known_groebner_bases = (table( [ ] )))](/support/helpjp/helpview.aspx?si=8505/file02595/math143.png)
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![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]] ] )))](/support/helpjp/helpview.aspx?si=8505/file02595/math150.png)
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![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]] ] )))](/support/helpjp/helpview.aspx?si=8505/file02595/math157.png)
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![POLYNOMIALIDEAL(x^2+3, y^2+1, z^2+2, variables = {x, y, z}, characteristic = 5, known_groebner_bases = (table( [ ] )))](/support/helpjp/helpview.aspx?si=8505/file02595/math164.png)
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![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]] ] )))](/support/helpjp/helpview.aspx?si=8505/file02595/math185.png)
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![POLYNOMIALIDEAL(y^2-z^2*x, z^2*x-2*x*y, variables = {x, y, z}, characteristic = 0, known_groebner_bases = (table( [ ] )))](/support/helpjp/helpview.aspx?si=8505/file02595/math192.png)
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![POLYNOMIALIDEAL(x+y, x^2+y^2-1, variables = {x, y}, characteristic = 0, known_groebner_bases = (table( [ ] )))](/support/helpjp/helpview.aspx?si=8505/file02595/math213.png)
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![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]] ] )))](/support/helpjp/helpview.aspx?si=8505/file02595/math220.png)
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![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]] ] )))](/support/helpjp/helpview.aspx?si=8505/file02595/math235.png)
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![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]] ] )))](/support/helpjp/helpview.aspx?si=8505/file02595/math242.png)
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References
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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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