D. Eisenbud, C. Huneke, and W. Vasconcelos. Direct methods for primary decomposition. Invent. Math., 110:207-235, 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... Huneke and Stillman about these ideals: is the doubly exponential behavior due to the number of minimal and or associated primes, or to the nature of one of them There exist algorithms for computing primary decompositions of ideals (see GianniTrager Zacharias [GTZ] Eisenbud Huneke Vasconcelos [EHV], or Shimoyama Yokoyama [SY] and they have been implemented on the symbolic computer algebra program Singular and partially on Macaulay2. However, the Mayr Meyer ideals have variable degree and a variable number of variables over an arbitrary field, and there are no algorithms to deal with this ....

D. Eisenbud, C. Huneke and W. Vasconcelos, Direct methods for primary decomposition, Invent. math., 110 (1992), 207-235.

....One way is to compute the sub variety H G(1; 3) where G(1; 3) denotes the Grassmannian of the lines of P 3 ) given by the lines contained in S. The computations of the primary decomposition of the ideal defining H (i.e. its zeros) with one of the known techniques (see for instance [GTZ] [EHV] and the references given there) allows us to determine the points of H. Clearly, a line of S is rational iff it is given by a point of H that can be expressed with rational coordinates. The computation of the variety H is straightforward: if F (x; y; z; t) 0 is the equation of S and if (p 1 ; ....

D. Eisenbud, C. Huneke, W. Vasconcelos, Direct Methods for Primary Decomposition, Invent. Math. 110 (1995), 207--235.

....are all rational: one way is to compute the sub variety H G(1; 3) where G(1; 3) denotes the Grassmannian of the lines of P 3 ) given by the lines contained in S. The computations of the primary decomposition of the ideal defining H with one of the known techniques (see for instance [GTZ] [EHV] and the references given there) allows us to determine the points of H . A line of S is rational iff it is given by a point of H that can be expressed with rational coordinates. The computation of the variety H is straightforward: if F (x; y; z; t) 0 is the equation of S and if (p 1 ; ....

D. Eisenbud, C. Huneke, W. Vasconcelos, Direct Methods for Primary Decomposition, Invent. Math. 110 (1995), 207--235

....I n (I n P kn ) which gives a desired primary decomposition of I n . For general ideals, it is very difficult to find the integer k which satisfies the theorem. For one thing, computing the embedded components of primary decompositions of arbitrary ideals is very difficult (see [EHV]) However, for monomial ideals in polynomial rings, computing primary decompositions is quite fast, see for example [STV] It turns out that one can obtain k for monomial ideals even without any primary decomposition algorithms: in [SS] Smith and I prove that if I and J are monomial ideals in the ....

D. Eisenbud, C. Huneke and W. Vasconcelos, Direct methods for primary decomposition, Invent. math., 110 (1992), 207-235.

....strictly contained in Sing(A) if we de ne J as above with c the minimal height of minimal primes of I . In this case we need another ideal. There are several alternatives to compute an ideal I Sing with V (I Sing ) Sing(A) Either we compute an equidimensional decomposition I = T i I i , [8, 15], of I , compute the Jacobian ideal J i for each equidimensional ideal I i and compute the ideal describing the intersection of any two equidimensional parts. The same works for a primary decomposition, 11, 8, 15] instead of an equidimensional decomposition. We can avoid an equidimensional, ....

....(I Sing ) Sing(A) Either we compute an equidimensional decomposition I = T i I i , 8, 15] of I , compute the Jacobian ideal J i for each equidimensional ideal I i and compute the ideal describing the intersection of any two equidimensional parts. The same works for a primary decomposition, [11, 8, 15]) instead of an equidimensional decomposition. We can avoid an equidimensional, resp. primary, decomposition if we compute the ideal of the non free locus of the module of K ahler di erentials, 1 A=K = 1 S=K . k X i=1 f i 1 S=K k X i=1 Sdf i ; where 1 S=K = L n ....

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Eisenbud D.; Huneke, C.; Vasconcelos, W.: Direct Methods for Primary Decomposition. Invent. Math. 110, 207-235 (1992).

....spread of I. A more familiar description is the one of I as the intersection of all prime ideals containing I or, equivalently, as the intersection of all minimal primes over I. It is well known that this intersection is finite. Also, by work of D. Eisenbud, C. Huneke and W.V. Vasconcelos [6], it is now easy to give an algorithmic approach to I suitable for effective computer calculations. On the other hand, reductions of an ideal are highly non unique. Their intersection, dubbed core of the ideal I, comes from a more recent vintage. It was studied for the first time by D. Rees ....

D. Eisenbud, C. Huneke and W.V. Vasconcelos, Direct methods for primary decomposition, Invent. Math. 110 (1992), 207-235.

.... ring normalisation, versal deformations, and many more (cf. GPS] The development of new algorithms provides, in addition, a better understanding and often even produces new theoretical insight, as has been the case, just to mention one example, for primary decomposition (cf. GTZ] [EHV]) This has also been the case for some topics treated in the present article. For more applications of computer algebra to algebraic geometry and singularity theory see [Gr2] We assume the reader is familiar with the main notions of Grobner bases (cf. CLO1] BW] I should like to thank T. ....

D. Eisenbud, C. Huneke and W. Vasconcelos, Direct methods for primary decomposition, Invent. Math., 110, 207--235 (1992).

....K(x 1 ; x d ) and to compute the radical p e I. Note that e I is zero dimensional. Therefore everything hinges on the feasibility of the computation of zero dimensional radicals over a rational function field over the original ground field. It should be mentioned that Eisenbud et al. [9] gave an algorithm for computing radicals which does not reduce to the zero dimensional case. However, the limitation of this algorithm is that it requires the ground field K to be of characteristic 0, or that K[x 1 ; x n ] I is generated by elements whose index of nilpotency is less than ....

....radicals which does not reduce to the zero dimensional case. However, the limitation of this algorithm is that it requires the ground field K to be of characteristic 0, or that K[x 1 ; x n ] I is generated by elements whose index of nilpotency is less than char(K) see Theorem 2. 7 in [9]) The aim of this paper is to present a new algorithm for the computation of zerodimensional radicals which works over any field K which is finitely generated over its prime field. For the case that K is perfect, there are good algorithm for this purpose given by Seidenberg [21] see also Becker ....

David Eisenbud, Craig Huneke, Wolmer V. Vasconcelos, Direct Methods for Primary Decomposition, Invent. Math. 110 (1992), 207--235.

....This method gives a good approximation of the primary decomposition of the variety [43, 58, 57] This is a straightforward method which gives cases described by equations and inequations. For each case sharp values of the dimensions can be determined. The method of equidimensional decomposition [15, 11], does not depend on factorization, and gives a decomposition of the radical of an ideal I generated by the given system in the form p I = I d1 1 I d2 2 : I dn n where each I d j j is an intersection of k j irreducible prime ideals of the same dimension d j . The number n, the ....

Eisenbud, D., Huneke, C. and Vasconcelos, W. 1992. Direct methods for primary decomposition, Invent. Math. 110 210--235.

....ideals. Thanks to Rosenfeld s lemma and its corollaries, proving the property amounts to applying the Jacobian criterion for regularity. The form of this commutative algebra result that we will use here has been applied in direct algorithms for the computation of primary decomposition of ideals (Eisenbud et al. 1992), Vasconcelos 1998) Theorem 3.2. Let A be a coherent autoreduced set of FfY g. A] H 1 A is a radical differential ideal. Proof. Note first that for any finitely generated ideal I and any f in a polynomial ring F [X] I : f 1 is equal to the intersection of those primary components of I ....

....3.2. Let A be a coherent autoreduced set of FfY g. A] H 1 A is a radical differential ideal. Proof. Note first that for any finitely generated ideal I and any f in a polynomial ring F [X] I : f 1 is equal to the intersection of those primary components of I with radical not containing f (Eisenbud et al. 1992, lemma 2.4) SA , the product of the separants of A, is the determinant of a maximal square submatrix of the Jacobian matrix of the set of polynomials A in the polynomial ring F [ Theta AY ] Thus SA belongs to the Jacobian ideal of (A) If 1 2 (A) H 1 A then [A] H 1 A = FfY g and the ....

Eisenbud, D., Huneke, C., Vasconcelos, W. (1992). Direct method for primary decomposition.

....whose zero set in GL(r; C) is the Galois group G of L. This proves 1. The map G G=G o induces an isomorphism of G=G o onto (G=R u ) G=Ru ) o . Therefore to prove 2. we shall show how to compute (G=R u ) G=R u ) o . Compute G=Ru as in 1. Standard arguments (Gianni et al. 1988; Eisenbud et al. 1992) allow one to decompose the variety defined by ff i = 0g into irreducible components. The number of these components will be j(G=R u ) G=R u ) o j. Selecting a distinct set of representatives fg j g from these components and deciding to which component each g i g j belongs allows us to write ....

Eisenbud, D., Huneke, C., Vasconcelos, W. (1992). Direct methods for primary decomposition. Inventiones Mathematicae, 110:207--236.

....algorithms for finding primary decompositions is an active area of research. In [FR] for example, the authors show that there exists a polynomial time algorithm for computing the primary decomposition of the ALGORITHMS FOR POLYCYCLIC BY FINITE MATRIX GROUPS 7 radical of a matrix algebra. In [EHV], the authors discuss the problem of finding a practical algorithm for computing the primary decomposition of an ideal in a polynomial ring. IBM s symbolic computation package AXIOM has a function for finding the primary decomposition for a zero dimensional ideal in a polynomial ring. Further ....

....primary decomposition for a zero dimensional ideal in a polynomial ring. Further research on practical algorithms for computing the primary decomposition will shed light on how best to achieve the reduction we seek here. One method is to mimic the proof of Proposition 4. 1, relying on methods in [EHV] or [FR] for deciding whether or not A is a field and, if not, for finding a zero divisor a in A. Now let us assume that G is an abelian subgroup of GL(n; R) given by a finite set of generators, and that we have found a basis for R n satisfying the criteria of Proposition 4.1. It is easy to see ....

D. Eisenbud, C. Huneke, W. Vasconcelos, Direct methods for primary decomposition, Invent. Math. 110 (1992), 207--235.

....algebra of G is a field. The development of practical algorithms for finding primary decompositions is an active area of research. In [10] for example, the authors show that there exists a polynomial time algorithm for computing the primary decomposition of the radical of a matrix algebra. In [9], the authors discuss the problem of finding a practical algorithm for computing the primary decomposition of an ideal in a polynomial ring. IBM s symbolic computation package AXIOM has a function for finding the primary decomposition for a zero dimensional ideal in a polynomial ring. Further ....

....primary decomposition for a zero dimensional ideal in a polynomial ring. Further research on practical algorithms for computing the primary decomposition will shed light on how best to achieve the reduction we seek here. One method is to mimic the proof of Proposition 2. 3, relying on methods in [9] or [10] for deciding whether or not A is a field and, if it is not, for finding a zero divisor a in A. Note that when R = Z, we find a basis for Z n relative to which G has the desired form. Now let us assume that G is an abelian subgroup of GL(n; R) given by a finite set fg 1 ; g k g ....

David Eisenbud, Craig Huneke, and Wolmer Vasconcelos. Direct methods for primary decomposition. Invent. Math., 110:207--235, 1992.

....into a union of finitely many irreducible and noncontractive components. The decomposition of an algebraic set is unique up to the order in which the components are written. Algorithms for finding the decomposition can be found in [Her26] Rit32, Rit50] MRR88] Sei74, Sei84] GTZ88] and [EHV90]. Among them, an efficient method for finding the decomposition is the method of Ritt s based on characteristic set computations (see [Rit50] Cho88] and [Wu94] Using Ritt s method, to decompose an algebraic set, we first compute a sequence of irreducible characteristic sets which has the union ....

D. Eisenbud, C. Huneke, and W. Vasconcelos. Direct methods for primary decomposition. Invent. Math., 1990.

.... ) 1 = J (i) Result = Result J (i) W = W; h (i) if this is done for i = 1; s, then Result = equidimensional radical of I if dimW dim I) Result = Result radical (W ) ffl return Result A quite different approach is due to Eisenbud, Huneke and Vasconcelos ([EHV]) We fix a field K of characteristic 0, or characteristic p, sufficiently large. Let A = K[x 1 ; x n ] I be a K algebra of finite type. We denote by J a (A) the a th fitting ideal of Omega AjK , the module of Kahler differentials and by J a (I) its pull back in K[x 1 ; x n ....

.... Delta Delta x n if and only if for a minimal Grobner basis G and G K[x n ] fgg we have dimK K[x 1 ; x n ] I = deg(g) and g is irreducible. Remark 2.7. An equivalent approach, also going to general position, is the following algorithm proposed by Eisenbud, Huneke, Vasconcelos ([EHV]) Algorithm 2.8. decomp EHV(I) Input: a zero dimensional radical ideal I Output: the associated prime ideals ffl Choose a generic f 2 K[x 1 ; x n ] and test whether f is a zero divisor mod I (this is the case if I : f I) If f is a zero divisor mod I (which implies I = Gamma ....

[Article contains additional citation context not shown here]

Eisenbud, D.; Huneke, C.; Vasconcelos, W.: Direct methods for primary decomposition. Invent. math. 110, 207--235 (1992).

....x 4 24 ; x 4 25 i) x 11 x 15 x 31 x 35 ) 3 (x 11 x 35 Gammax 15 x 31 ) The associated prime is the monomial prime hx 12 ; x 13 ; x 14 ; x 32 ; x 33 ; x 34 ; x 21 ; x 22 ; x 23 ; x 24 ; x 25 i. We note that one can check that all the ideals above are primary by either using Theorem 1. 1 in [4] or Algorithm 9.4 in [5] We summarize this in the following. Proposition 3.7 The ideal J 35 has 19 associated primes, 10 minimal and 9 embedded. A minimal primary decomposition is given by J 35 = rad(J 35 ) G 1 G 2 H K 1 K 2 L R 1 R 2 U: The next figure is the partially ....

D. Eisenbud, C. Huneke and W. Vasconcelos, Direct methods for primary decomposition, Invent. math. 110 (1992) pp. 207--235.

.... First of all, we wish to contribute to a better geometric understanding of flatness, together with classical and recent papers on the matter (see e.g. 3] 4] 7] 9] 10] Secondly, at least for affine algebraic maps, our result is interesting from an algorithmic point of view (see e.g. [5] and [3] as it establishes a link between flatness and primary decomposition via the simple algorithmic construction of the n th tensor power. Auslander s proof [1] relies on properties, specific to finitely generated modules, which are no longer true in our setup. Nevertheless, we will follow ....

D.Eisenbud, C.Huneke, W.V.Vasconcelos, Direct methods for primary decomposition, Invent. Math. 110 (1992), pp. 207-235.

....algebra of G is a field. The development of practical algorithms for finding primary decompositions is an active area of research. In [FR] for example, the authors show that there exists a polynomial time algorithm for computing the primary decomposition of the radical of a matrix algebra. In [EHV], the authors discuss the problem of finding a practical algorithm for computing the primary decomposition of an ideal in a polynomial ring. IBM s symbolic computation package AXIOM has a function for finding the primary decomposition for a zero dimensional ideal in a polynomial ring. Further ....

....decomposition for a zero dimensional ideal in a polynomial ring. Further research on practical algorithms for computing the primary decomposition will shed light on how best to achieve the reduction we seek here. One method is to mimic the proof of 17 Proposition 2.4. 1, relying on methods in [EHV] or [FR] for deciding whether or not A is a field and, if it is not, for finding a zero divisor a in A. Note that when R = Z, we find a basis for Z n relative to which G has the desired form. Now let us assume that G is an abelian subgroup of GL(n; R) given by a finite set fg 1 ; g k g ....

[Article contains additional citation context not shown here]

D. Eisenbud, C. Huneke, W. Vasconcelos, Direct methods for primary decomposition, Invent. Math. 110 (1992), 207--235.

....of the Mayr Meyer ideals must be due to the embedded prime ideals. The structure of the embedded prime ideals of the Mayr Meyer ideals is examined in [S2] There exist algorithms for computing primary decompositions of ideals (see GianniTrager Zacharias [GTZ] Eisenbud Huneke Vasconcelos [EHV], or Shimoyama Yokoyama [SY] and they have been partially implemented on the symbolic computer algebra programs Singular [GPS] and Macaulay2 [GS] However, the Mayr Meyer ideals have variable degree and a variable number of variables over an arbitrary field, and there are no algorithms to deal ....

D. Eisenbud, C. Huneke and W. Vasconcelos, Direct methods for primary decomposition, Invent. math., 110 (1992), 207-235.

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D. Eisenbud, C. Huneke, and W. Vasconcelos. Direct methods for primary decomposition. Invent. Math., 110:207-235, 1992.

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D. Eisenbud, C. Huneke, and W. Vasconcelos. Direct methods for primary decomposition. Inventiones Mathematicae, 110:207-235, 1992.

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Eisenbud, D., Huneke, C., and Vasconcelos, W., Direct methods for primary decomposition. Inv. Math. 110 (1992) 207-235

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D. Eisenbud, C. Huneke, W. Vasconcelos (1992): Direct methods for primary decomposition, Invent. Math. 110 207--235.

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D. Eisenbud, C. Huneke and W. Vasconcelos, Direct methods for primary decomposition, Invent. Math. 110 (1992), 207-235.

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D. Eisenbud, C. Huneke, W. Vasconcelos, Direct methods for primary decomposition, Invent. Math. 110 (1992), 207-235

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