We present an algorithm to compute the primary decomposition of a submodule N of the free module Z[x1, . . . , xn]. For this purpose we use algorithms for primary decomposition of ideals in the polynomial ring over the integers. The idea is to compute first the minimal associated primes of N , i.e. the minimal associated primes of the ideal Ann(Z[x1, . . . , xn]/N ) in Z[x1, . . . , xn] and the...

The rst purpose of this paper is to describe a new mathematical approach for the computation of an irredundant primary decomposition of a given polynomial ideal I. This presentation will be formed of three parts : a decomposition of the associated radical ideal p I to an intersection of prime ideals P i , then the determination of ideals I i whose the radical is prime (equal to P i), and lastly...

Grete Hermann proved in H] that for any ideal I in an n-dimensional polynomial ring over the eld of rational numbers, if I is generated by polynomials f 1 ; : : : ; f k of degree at most d, then it is possible to write f = P r i f i such that each r i has degree at most deg f + (kd) (2 n). Mayr and Meyer in MM] found (generators) of ideals for which a doubly exponential bound in n is indeed ach...

This paper presents a new algorithm that computes the local algebras of the roots of a zero-dimensional polynomial equation system, with a number of operations in the coefficient field that is polynomial in the number of variables, in the evaluation cost of the equations and in a Bézout number.

Consider the discrete conditional independence model M given by {X1 ⊥ X2 | X3, X1 ⊥ X3 | X2}. The intersection axiom for conditional independence can be applied with the statements of M as premises to derive the conclusion X1 ⊥ (X2, X3). But the independence axiom only holds in general when X is in the interior of the probability simplex, and it’s a natural question to ask what can be inferred ...

Consider an ideal I ⊂ R = C[x] = C[x 1 ,. .. , x n ] defining a complex affine variety X = V (I) = {x ∈ C n | ∀f ∈ I, f (x) = 0}. We describe the associated components VAss(I) = {V (P) | P ∈ Ass(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) = V (I (d)) in a higher-dimensional comple...

We study global primary decompositions in the category of sheaves on a scheme which are equivariant under the action of an algebraic group. We show that equivariant primary decompositions exist if the group is connected. As main application we consider the case of toric sheaves over toric varieties. Comparing these decompositions with primary decompositions of graded and fine graded modules ove...

We prove that each ideal of a locally formally equidimensional analytically un-ramiied Noetherian integral domain is the contraction of an ideal of a one-dimensional semilo-cal birational extension domain. We give an application to a problem concerning the primary decomposition of powers of ideals in Noetherian rings. It is shown in S2] that for each ideal I in a Noetherian commutative ring R t...