An algorithm for primary decomposition in polynomial rings over the integers

Pfister, Gerhard; Sadiq, Afshan; Steidel, Stefan

doi:10.2478/s11533-011-0037-8

Cited by 5 publications

(5 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…, x n ] date back to 1978 [4,30]. More recently, Pfister et al [27] presented a slightly different approach. Inspired by this algorithm, we gave an efficient algorithm in [20] for computing the primary decomposition of ideals I ⊆ Z[x 1 , .…”

Section: Computing the Associated Primes Of Finite Z-algebrasmentioning

confidence: 99%

“…In this case we will also have to factor (potentially large) integers, as already the example R = Z/nZ shows. Since the 1970s, various approaches have been taken to tackle these tasks, starting with the case of an algebra R which is a finitely generated Z-module (see [2], [4], [30], [27]). At the core of most of these algorithms lies the calculation of strong Gröbner bases for ideals in Z[x 1 , .…”

Section: Introductionmentioning

confidence: 99%

“…In Section 4 we then use the ideas of [27] to compute the associated primes of an explicitly given finite Z-algebra R (see Algorithm 4.2). The method is to distinguish between the prime ideals which contain an integer prime number and those which don't.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Efficient Algorithms for Finite $\mathbb{Z}$-Algebras

Kreuzer,

Walsh

2024

Journal of Groups, Complexity, Cryptology

View full text Add to dashboard Cite

For a finite $\mathbb{Z}$-algebra $R$, i.e., for a $\mathbb{Z}$-algebra which is a finitely generated $\mathbb{Z}$-module, we assume that $R$ is explicitly given by a system of $\mathbb{Z}$-module generators $G$, its relation module ${\rm Syz}(G)$, and the structure constants of the multiplication in $R$. In this setting we develop and analyze efficient algorithms for computing essential information about $R$. First we provide polynomial time algorithms for solving linear systems of equations over $R$ and for basic ideal-theoretic operations in $R$. Then we develop ZPP (zero-error probabilitic polynomial time) algorithms to compute the nilradical and the maximal ideals of 0-dimensional affine algebras $K[x_1,\dots,x_n]/I$ with $K=\mathbb{Q}$ or $K=\mathbb{F}_p$. The task of finding the associated primes of a finite $\mathbb{Z}$-algebra $R$ is reduced to these cases and solved in ZPPIF (ZPP plus one integer factorization). With the same complexity, we calculate the connected components of the set of minimal associated primes ${\rm minPrimes}(R)$ and then the primitive idempotents of $R$. Finally, we prove that knowing an explicit representation of $R$ is polynomial time equivalent to knowing a strong Gr\"obner basis of an ideal $I$ such that $R = \mathbb{Z}[x_1,\dots,x_n]/I$.

show abstract

Section: Computing the Associated Primes Of Finite Z-algebrasmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Efficient Algorithms for Finite $\mathbb{Z}$-Algebras

Kreuzer,

Walsh

2024

Journal of Groups, Complexity, Cryptology

View full text Add to dashboard Cite

show abstract

“…Algorithms for primary decomposition of polynomial ideals and decomposition of an affine variety into a union of irreducible components are provided by [1], [6], [5], [18], [15], [10], [13] and other sources.…”

Section: Tangent Bundle Interpolation Of a Finite Family Of Codesmentioning

confidence: 99%

“…(ii) Let X/F q ⊂ F q n and Y /F q ⊆ F q n be irreducible affine varieties, defined over F q . Then the direct sum T a (X, F q δ(a,b) )⊕T b (Y, F q δ(a,b) ) = T (a,b) (X ×Y, F q δ(a,b) ) at ∀(a, b) ∈ X smooth ×Y smooth (13) of tangent spaces to X and Y is a tangent space to X × Y . (iii) Let X/F q ⊂ F q n be an irreducible affine variety, defined over F q and g = (g 1 , .…”

Section: 3mentioning

confidence: 99%

Tangent Codes

Kasparian¹,

Velikova²

2014

Preprint

View full text Add to dashboard Cite

The present article studies the finite Zariski tangent spaces to an affine variety X as linear codes, in order to characterize their typical or exceptional properties by global geometric conditions on X. The discussion concerns the generic minimum distance of a tangent code to X, its lower semi-continuity under a deformation of X, as well as the existence of Zariski tangent spaces to X with exceptional minimum distance. Tangent codes are shown to admit simultaneous decoding. The duals of the tangent codes to X are realized by gradients of polynomials from the ideal of X. We provide constructions of affine varieties with near MDS, cyclic or Hamming tangent codes. Puncturing, shortening and extending finite Zariski tangent spaces are related to the corresponding operations on affine varieties. The (u|u + v) construction of tangent codes is associated with a fibered product of varieties. Explicit constructions realize linear Hamming isometries as differentials of morphisms of affine varieties.

show abstract