This page describes the Magma implementation of the algorithms described in [2]. Also included is an implementation of algorithm ZPDF in [1].

The implementation is available here: primarydecomposition.tgz.

To use the algorithm, after starting magma, type

> Attach("primarydecomposition.magma");This provides three methods: radical (Algorithm 4.8 in [2]), primaryDecompositionInfinite (Algorithm 5.16 in [2]), and primaryDecomposition (Algorithm 6.5 in [2]).

To use the algorithm ZPDF, after starting magma, type

> Attach("primarydecomposition.magma"); > Attach("gtz.magma");This provides the additional method primaryDecompositionGTZ (an implementation of Algorithm ZPDF in [1]).

All four methods take as argument a zero-dimensional ideal I with computed Gröbner basis. The method radical computes the radical of I, and the other methods compute the primary decomposition of I.

> R< x, y > := PolynomialRing(Rationals(), 2, "grevlex"); > I := ideal< R | (x^2 + 1)^2, y^2 + 1 >; > Groebner(I); > radical(I); Ideal of Polynomial ring of rank 2 over Rational Field Order: Graded Reverse Lexicographical Variables: x, y Inhomogeneous, Dimension 0 Groebner basis: [ x^2 + 1, y^2 + 1 ] > primaryDecomposition(I); [ Ideal of Polynomial ring of rank 2 over Rational Field Order: Graded Reverse Lexicographical Variables: x, y Inhomogeneous, Dimension 0 Groebner basis: [ x^2 - 2*x*y - 1, y^2 + 1 ], Ideal of Polynomial ring of rank 2 over Rational Field Order: Graded Reverse Lexicographical Variables: x, y Inhomogeneous, Dimension 0 Groebner basis: [ x^2 + 2*x*y - 1, y^2 + 1 ] ] > primaryDecompositionInfinite(I); [ Ideal of Polynomial ring of rank 2 over Rational Field Order: Graded Reverse Lexicographical Variables: x, y Inhomogeneous, Dimension 0 Groebner basis: [ x^2 + 2*x*y - 1, y^2 + 1 ], Ideal of Polynomial ring of rank 2 over Rational Field Order: Graded Reverse Lexicographical Variables: x, y Inhomogeneous, Dimension 0 Groebner basis: [ x^2 - 2*x*y - 1, y^2 + 1 ] ] > primaryDecompositionGTZ(I); [ Ideal of Polynomial ring of rank 2 over Rational Field Order: Graded Reverse Lexicographical Variables: x, y Inhomogeneous, Dimension 0 Groebner basis: [ x^2 + 2*x*y - 1, y^2 + 1 ], Ideal of Polynomial ring of rank 2 over Rational Field Order: Graded Reverse Lexicographical Variables: x, y Inhomogeneous, Dimension 0 Groebner basis: [ x^2 - 2*x*y - 1, y^2 + 1 ] ]

finite1_ilias13.magma finite2_Reimer_5a.magma finite3_Steidel_1.magma rationals1.magma rationals2.magma rationals3.magma functionfield1.magma functionfield2.magma functionfield3.magmaAfter attaching the packages primarydecomposition.magma and gtz.magma as described above, the benchmarks can be run as follows.

> load "finite1_ilias13.magma"; Loading "finite1_ilias13.magma" > time pd := primaryDecomposition(I); Time: 2.160 > time pdgtz := primaryDecompositionGTZ(I); Time: 3.400

[1] |
P. Gianni, B. Trager, and G. Zacharias Gröbner bases and primary decomposition of polynomial ideals J. Symbolic Comput. 6 (1988), no. 2-3, 139-167. |

[2] |
S. Jambor Primary decomposition of zero-dimensional ideals over arbitrary fields Preprint (2014). |