. A G ] 2 9 M ay 2 00 8 NUMERICAL PRIMARY DECOMPOSITION ANTON
نویسنده
چکیده
Consider an ideal I ⊂ R = C[x 1 ,. .. , x n ] defining a complex affine variety X ⊂ C n. We describe the components associated to I by means of numerical primary decomposition (NPD). The method is based on the construction of deflation ideal I (d) that defines the deflated variety X (d) in a complex space of higher dimension. For every embedded component there exists d and an isolated component Y (d) of I (d) projecting onto Y. In turn, Y (d) can be discovered by existing methods for prime decomposition, in particular, the numerical irreducible decomposition, applied to X (d). The concept of NPD gives a full description of the scheme Spec(R/I) by representing each component with a witness set. We propose an algorithm to produce a collection of witness sets that contains a NPD and that can be used to solve the ideal membership problem for I.
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