# Local Decomposition Algorithms

@inproceedings{Alonso1990LocalDA, title={Local Decomposition Algorithms}, author={Maria Emilia Alonso and Teo Mora and Mario Di Raimondo}, booktitle={AAECC}, year={1990} }

INTRODUCTION For an ideal I ⊂ P := k[X 1 ,…,X n ], many algorithms using Gröbner techniques are a direct consequence of the definition itself of Gröbner basis: among them we can list algorithms for computing syzygies, dimension, minimal bases, free resolutions, Hilbert function and other algebro-geometric invariants of an ideal; the explicit knowledge of syzygies allows moreover to compute ideal intersection and quotients. Other algorithms require a specific property of Gröbner bases w.r.t. of…

## 26 Citations

### Algorithms in Local Algebra

- MathematicsJ. Symb. Comput.
- 1995

This work uses the generalization of Mora's tangent cone algorithm to arbitrary term orders to give a detailed description of the necessary modifications and restrictions for term orders of mixed type.

### Algorithms in Local Algebra

- Mathematics
- 1995

Let k be a eld, S = k[xv : v 2 V ] be the polynomial ring over the nite set of variables (xv : v 2 V), and m = (x v : v 2 V) the ideal dening the origin of Spec S. It is theoretically known (see e.g.…

### Computing generators of the ideal of a smooth affine algebraic variety

- Mathematics, Computer ScienceJ. Symb. Comput.
- 2004

### Effective Computation of Radical of Ideals and Its Application to Invariant Theory

- Computer Science, MathematicsICMS
- 2014

This paper introduces a new notion of genericity and exhibits a novel deterministic algorithm to put an ideal in Nœther position and uses this algorithm and also the algorithm due to Krick and Logar to present an efficient algorithm to calculate the radical of a complete intersection ideal.

### Extractions: Computable and Visible Analogues of Localizations for Polynomial Ideals

- MathematicsArXiv
- 2015

It is proved that extractions are as powerful as localizations in the sense that for any multiplicatively closed subset of $S$ of $K[x_1,\ldots,x_n]$ and any polynomial ideal $I$, there always exists a polynometric ideal $J$ such that $\beta(I,J)=(S^{-1}I)^c$.

### Minimal Primary Decomposition and Factorized Gröbner Bases

- Computer ScienceApplicable Algebra in Engineering, Communication and Computing
- 1997

A way to interweave factorized Gröbner bases and the ideas in [5] that leads to a significant speed up in the computation of isolated primes for well splitting examples is described and a method to detect necessary embedded primes in the output collection of the algorithm is outlined.

### On the Computation of the Radical of Polynomial Complete Intersection Ideals

- MathematicsAAECC
- 1995

A single exponential algorithm is exhibited which computes a system of generators of certain polynomial ideals with radical d:=maxj deg fj, the generated ideal and its radical.

### Constructive Ideal Theory

- Mathematics
- 2015

This chapter introduces some algorithms of constructive ideal theory, almost all of which are based on Grobner bases, and provides the basic algorithmic tools which will be used in later chapters.

### Decision of Algebra Isomorphisms Using Gröbner Bases

- Mathematics
- 1993

Given two finite-dimensional non-commutative finitely presented algebras A and B over a field k, this paper presents two algorithms for deciding whether or not they are isomorphic as k-algebras. We…

### Derivations and Radicals of Polynomial Ideals over Fields of Arbitrary Characteristic

- Mathematics, Computer ScienceJ. Symb. Comput.
- 2002

A complete effective solution to the problem of computing radicals of polynomial ideals over general fields of arbitrary characteristic by using derivations and ideal quotients to recover as much of the radical as possible.

## References

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Borders are given for the complexity of the above problems which are simply exponential in the number n of variables in the one-dimensional case, and it is shown that in the general case the first two problems are doubly exponential only in the dimension of the ideal.

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- Mathematics, Computer ScienceJ. Symb. Comput.
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- 1989