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An algorithm for the computation of the radical of an ideal

2006
*
Proceedings of the 2006 international symposium on Symbolic and algebraic computation - ISSAC '06
*

. , x n ] • I

doi:10.1145/1145768.1145802
dblp:conf/issac/Laplagne06
fatcat:lhtuvgyfnrf6rcbj4zsmitcox4
*ideal*in k[x]*The**radical**of**an**ideal*√ I = {f ∈ k[x] / f m ∈ I*for*some m ∈ N} • V(I) = V( √ I). ...,x n ,t] with t a new variable. • √ I ∩ J = √ I ∩ √ J. ...*Radical*membership f ∈ √ I ⇐⇒ 1 ∈ I, tf − 1 k[x 1 , Primary decomposition Every*ideal*I ⊂ k[x] can be decomposed as*an*intersection I = Q 1 ∩ · · · ∩ Q t*of*primary*ideals*, with √ Q i = P i prime. ...##
###
Prime Decompositions of Radicals in Polynomial Rings

1994
*
Journal of symbolic computation
*

We show that prime decomposition

doi:10.1006/jsco.1994.1052
fatcat:36lszgijy5hnhib56ckk4swolq
*algorithms*in R can be lifted to R[x] if*for*every prime*ideal*P in R univariate polynomials can be factored over*the*quotient field*of**the*residue class ring R/P . ... In this paper we are concerned with*the**computation**of*prime decompositions*of**radicals*in polynomial rings over a noetherian commutative ring R with identity. ... Acknowledgement: I want to thank both referees*for*their detailed and helpful comments on*an*earlier version*of*this paper. ...##
###
Computing the Radical of an Ideal in Positive Characteristic

2001
*
Journal of symbolic computation
*

We propose a method

doi:10.1006/jsco.2001.0446
fatcat:2kvlehe43jghran32d2b7yslhe
*for**computing**the**radical**of**an*arbitrary*ideal*in*the*polynomial ring in n variables over a perfect field*of*characteristic p > 0. ...*of*generators*of**the**ideal*and 3. ... Kazuhiro Yokoyama*for*stimulating discussions,*the*anonymous referees*for*helpful comments, in particular,*for*providing*the*procedure given in Figure 1 , and Mr Konstantinos Slavakis*for*pointing out ...##
###
On the generalized Ritt problem as a computational problem

2009
*
Journal of Mathematical Sciences
*

*The*technique used in

*the*proof

*of*equivalence yields

*algorithms*

*for*

*computing*a canonical decomposition

*of*a

*radical*differential

*ideal*into prime components and a canonical generating set

*of*a

*radical*...

*The*Ritt problem asks if there is

*an*

*algorithm*that tells whether one prime differential

*ideal*is contained in another one if both are given by their characteristic sets. ... Acknowledgemenets We thank Michael Singer and William Sit

*for*their helpful suggestions and support, and Russell Miller

*for*

*the*introduction to

*the*theory

*of*

*computable*fields and subsequent discussions ...

##
###
Computing Real Radicals by Moment Optimization

2021
*
Proceedings of the 2021 on International Symposium on Symbolic and Algebraic Computation
*

We present a new

doi:10.1145/3452143.3465541
fatcat:x53tksj3w5dxzn44h2nadtynju
*algorithm**for**computing**the*real*radical**of**an**ideal*and, more generally,*the*-*radical**of*, which is based on convex moment optimization. ... We give*an*effective, general stopping criterion on*the*degree to detect when*the*prime*ideals*lying over*the*annihilator are real and*compute**the*real*radical*as*the*intersection*of*real prime*ideals*... ACKNOWLEDGMENTS*The*authors would like to thank*the*anonymous referees*for*their helpful suggestions. ...##
###
Computing real radicals by moment optimization
[article]

2021
*
arXiv
*
pre-print

We present a new

arXiv:2102.09367v1
fatcat:3hcz4twicjenli4oq7ia5uuvei
*algorithm**for**computing**the*real*radical**of**an**ideal*and, more generally,*the*-*radical**of*, which is based on convex moment optimization. ... We give*an*e ective, general stopping criterion on*the*degree to detect when*the*prime*ideals*lying over*the*annihilator are real and*compute**the*real*radical*as*the*intersection*of*real prime*ideals*lying ... ACKNOWLEDGMENTS*The*authors would like to thank*the*anonymous referees*for*their helpful suggestions. ...##
###
Intrinsic factorization of ideals in Dedekind domains
[article]

2018
*
arXiv
*
pre-print

We present a generalization

arXiv:1805.02289v1
fatcat:5ubiz36jajghdbpzyxpki3osmy
*of*a polynomial factorization*algorithm*that works with*ideals*in maximal orders*of*global function fields. ...*The*method presented in this paper is intrinsic in*the*sense that it does not depend on*the*embedding*of**the*ring*of*polynomials into*the*Dedekind domain in question. ... Thus,*the**radical**of*a is*computable*, as well by*the*previous lemma. We are now ready to present*an**algorithm**for**the**radical*decomposition. ...##
###
The Calculation of Radical Ideals in Positive Characteristic

2002
*
Journal of symbolic computation
*

We propose

doi:10.1006/jsco.2002.0560
fatcat:rlmmzwb3cnei7pjced5ejj4zz4
*an**algorithm**for**computing**the**radical**of*a polynomial*ideal*in positive characteristic.*The**algorithm*does not involve polynomial factorization. ... I thank Gerhard Pfister, Lorenzo Robbiano, and Wolmer Vasconcelos*for*helpful conversations and*for*their comments on*the*preprint version*of*this note. ... I am also thankful to*the*anonymous referees*for*suggesting improvements and pointing out some typing errors. ...##
###
Page 2855 of Mathematical Reviews Vol. , Issue 94e
[page]

1994
*
Mathematical Reviews
*

2855
94e:68089 68Q40 13P10
Krick, Teresa (RA-CONY; Buenos Aires);
Logar, Alessandro (I-TRST; Trieste)

*An**algorithm**for**the**computation**of**the**radical**of**an**ideal*in*the*ring*of*polynomials. ... “Finally, we recall that a similar solution*for**the**computation**of**the**radical**of**an**ideal*was presented in a paper by R. ...##
###
Page 6822 of Mathematical Reviews Vol. , Issue 98K
[page]

1998
*
Mathematical Reviews
*

One

*of**the**radicals*studied is*the*L-*radical**of**an**ideal*J*of*K[x] with respect to*an*arbitrary field extension L*of*K; it is equal to*the*vanishing*ideal**of**the*set*of*L-rational points*of*7. ...*Algorithms**for**radical**computations*and root counting design*for*binomial*ideals*are presented, and some Bezout-type bounds*for**the*number*of*L- rational points are given, in case their number is finite ...##
###
Local decomposition algorithms
[chapter]

1991
*
Lecture Notes in Computer Science
*

We propose here a new

doi:10.1007/3-540-54195-0_52
fatcat:hxawqvmtarh3rlbibw3cayfj74
*algorithm**for**radical**computation*; it is a variant*of**the**algorithm*in [GH] which doesn't require generic position. ... Genova*For**an**ideal*I ⊂ P := k[X 1 ,... ...*the**algorithm*above, followed by a test whether I = Top(I) g)*radical**computation*We don't know*of*any*algorithm**for**radical**computation*which doesn't pass through*the**computation**of**the*equidimensional ...##
###
A Divide and Conquer Method to Compute Binomial Ideals
[chapter]

2014
*
Lecture Notes in Computer Science
*

This work is a generalization

doi:10.1007/978-3-642-54423-1_56
fatcat:wframq5albavfnqtowl6m7usny
*of**the*work done by*the*authors in [12, 13] and is motivated by*the*fact that any*algorithm*to*compute*binomial*ideals*spends a significant amount*of*time*computing*Gröbner ... We apply*the*framework on five problems -*radical*, saturation, cellular decomposition, minimal primes*of*binomial*ideals*, and*computing*a generating set*of*a toric*ideal*. ...*For**an**ideal*I ⊆ R, √ I = { r | r m ∈ I, m ≥ 0 } is*the**radical**of*I. I : r ∞ = { s | sr j ∈ I,*for*some j ≥ 0 } will denote*the*saturation*of*I w.r.t. r. ...##
###
On Computation of Kolchin Characteristic Sets: Ordinary and Partial Cases
[article]

2006
*
arXiv
*
pre-print

In

arXiv:math/0606124v2
fatcat:reesxlqwcjeh3icdfq2oeauqpy
*the*partial differential case we give*an**algorithm**for**computing*characteristic sets in*the*special case*of**radical*differential*ideals*satisfying*the*property*of*consistency. ... In this paper we study*the*problem*of**computing*a Kolchin characteristic set*of*a*radical*differential*ideal*. ... We also thank referees*of**the*ACA 2004*for*their important remarks. ...##
###
The Normalization: a new Algorithm, Implementation and Comparisons
[chapter]

1999
*
Computational Methods for Representations of Groups and Algebras
*

Introduction We present a new

doi:10.1007/978-3-0348-8716-8_9
fatcat:odozkqznnjdpdnqu2fsvcmixoq
*algorithm**for**computing**the*normalization R*of*a reduced a ne ring R, together with some remarks on e ciency based on our experience with*an*implementation*of*this*algorithm*...*For*this situation there is*an*easy*algorithm*: Input: I K x 1 ; : : : ; x n ] a (weighted) homogeneous*radical**ideal*, deg(x 1 ) = = deg(x k ) = 0; deg(x i ) > 0*for*i > k, I \ K x 1 ; : : : ; x k ] being ...##
###
Algorithmic Properties of Polynomial Rings

1998
*
Journal of symbolic computation
*

In this paper we investigate how

doi:10.1006/jsco.1998.0227
fatcat:lmw5uzaf6zddxfcsfb7mxfl2ai
*algorithms**for**computing*heights,*radicals*, unmixed and primary decompositions*of**ideals*can be lifted from a Noetherian commutative ring R to polynomial rings over R. ... We say that*radicals*are*computable*in R if there exists*an**algorithm**radical*R which*computes**for*every finite subset F*of*R a basis*of**the**radical**of*F . ... In this paper we study whether this is also true*for**the**algorithmic*properties (1) heights*of**ideals*are*computable*, (2)*radicals**of**ideals*are*computable*, (3) unmixed decompositions*of**ideals*are*computable*...
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