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Yet another ideal decomposition algorithm [chapter]

Massimo Caboara, Pasqualina Conti, Carlo Traverse
1997 Lecture Notes in Computer Science  
Conclusion of the Algorithm for the Reduced Decomposition.  ...  Another possible reduction test uses jacobians.  ... 
doi:10.1007/3-540-63163-1_4 fatcat:odoiil4ztbc3hnq2oqbgzmprsm

Prime Decompositions of Radicals in Polynomial Rings

Michael Kalkbrener
1994 Journal of symbolic computation  
We show that prime decomposition 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 the proof of this result a lifting algorithm is constructed which can be considered as a generalization of the algorithm of Ritt and Wu.  ...  Another method based on a similar strategy is the algorithm for computing equidimensional decompositions of varieties in Kalkbrener (1993) . Compare also with Lazard (1991) and Wang (1993) .  ... 
doi:10.1006/jsco.1994.1052 fatcat:36lszgijy5hnhib56ckk4swolq

Page 1573 of Mathematical Reviews Vol. , Issue 99c [page]

1991 Mathematical Reviews  
Franz Pauer (Innsbruck) 99¢:13051 13P10 68Q40 Caboara, Massimo (I-PISA; Pisa); Conti, Pasqualina (I-PISA-DA; Pisa); Traverso, Carlo (I-PISA; Pisa) Yet another ideal decomposition algorithm.  ...  ), and the primary or the irreducible reduced decomposition can be more easily obtained from a pure-dimensional decomposition.  ... 

On the generalized Ritt problem as a computational problem

O. D. Golubitsky, M. V. Kondratieva, A. I. Ovchinnikov
2009 Journal of Mathematical Sciences  
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.  ...  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  ...  Then in the above decomposition we can remove all redundant components, i.e., those P i that contain another P j .  ... 
doi:10.1007/s10958-009-9689-3 fatcat:4gv3j2u3crgk7gm5izny2a4zpu

Triangular systems and factorized Gröbner bases [chapter]

Hans-Gert Gräbe
1995 Lecture Notes in Computer Science  
Such a refinement guarantees, different to the usual Gröbner factorizer, to produce a quasi prime decomposition, i.e. the resulting components are at least pure dimensional radical ideals.  ...  Triangular systems are a very helpful tool between factorization at a heuristical level and full decomposition into prime components.  ...  Below we present a quasi prime decomposition algorithm.  ... 
doi:10.1007/3-540-60114-7_18 fatcat:dhezik5bnnhgtlamqek54rshza

An algorithm for the Euclidean cell decomposition of a cusped strictly convex projective surface [article]

Stephan Tillmann, Sampson Wong
2015 arXiv   pre-print
We show that Weeks' algorithm to compute this decomposition for a hyperbolic surface generalises to strictly convex projective surfaces.  ...  Cooper and Long generalised Epstein and Penner's Euclidean cell decomposition of cusped hyperbolic manifolds of finite volume to non-compact strictly convex projective manifolds of finite volume.  ...  Introduction The Epstein-Penner decomposition [2] is an elegant yet powerful construction in the study of non-compact hyperbolic manifolds of finite volume.  ... 
arXiv:1512.01645v1 fatcat:uto3cw7bjndyfhuwwjnl6wfo6i

Efficient parallelization of molecular dynamics simulations with short-ranged forces

Ralf Meyer
2014 Journal of Physics, Conference Series  
The results show that the new algorithm gives consistent speedups which are depending on the properties of the simulated system either comparable or superior to those obtained with spatial decomposition  ...  In this work, this algorithm is tested on a variety of multi-core systems using three types of benchmark simulations.  ...  This system is very homogeneous and therefore nearly ideal for the spatial decomposition method.  ... 
doi:10.1088/1742-6596/540/1/012006 fatcat:n3shy55lqrd5jjbmvhirgn4sq4

Thirty years of Polynomial System Solving, and now?

Daniel Lazard
2009 Journal of symbolic computation  
decomposition seems to be less frequently needed, maybe because the ideals coming from applications are usually prime (in the above example of the implicitization, the ideal obtained by the saturation  ...  On the other hand, when solving a system, it is a priori unknown if the solution set is irreducible, and its decomposition in irreducible components is a difficult task, not yet well solved; therefore  ... 
doi:10.1016/j.jsc.2008.03.004 fatcat:umelra7aarag3fcsehfu67is6u

On Computation of Kolchin Characteristic Sets: Ordinary and Partial Cases [article]

Marina Kondratieva, Alexey Ovchinnikov
2006 arXiv   pre-print
We also discuss the usefulness of regular and characteristic decompositions of radical differential ideals.  ...  In 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.  ...  Denote [9, Algorithm 7.1] by χ-Decomposition.  ... 
arXiv:math/0606124v2 fatcat:reesxlqwcjeh3icdfq2oeauqpy

Computing strong regular characteristic pairs with Groebner bases [article]

Rina Dong, Dongming Wang
2020 arXiv   pre-print
We present some properties about strong regular characteristic pairs and characteristic decomposition and illustrate the proposed algorithm and its performance by examples and experimental results.  ...  Based on this strategy of splitting by means of quotient and with Groebner basis and ideal computations, we devise a simple algorithm to decompose an arbitrary polynomial set F into finitely many strong  ...  The W-characteristic set of an ideal may be morbid and it is not yet clear when morbidity happens.  ... 
arXiv:1907.13537v2 fatcat:4rlktjog7jfjdc726vlgn3vtua

Using Multi-Objective Bat Algorithm for Solving Multi-Objective Non-linear Programming Problem

Rajwan Hamood Sheah, Iraq T. Abbas
2021 Iraqi Journal of Science  
MOBATD is a multi-objective bat algorithm that incorporates the dominance concept with the decomposition approach.  ...  Decomposition is a basic strategy in traditional multi-objective optimization. However, it has not yet been widely used in multi-objective evolutionary optimization.  ...  In multi-objective enhancement, for the most part, there is no single ideal arrangement, yet rather a lot of Pareto ideal arrangements.  ... 
doi:10.24996/ijs.2021.62.3.29 fatcat:tegl5lf66zedvbz7p4d3ihfl2u

Block Stanley decompositions I. Elementary and gnomon decompositions

James Murdock, Theodore Murdock
2015 Journal of Pure and Applied Algebra  
We give two algorithms that generate different block decompositions, which we call elementary and gnomon decompositions, and give examples.  ...  In a sequel to this paper we will introduce two additional algorithms that generate block decompositions that may not always be subprime, but are always incompressible.  ...  We have not yet found an algorithm that always produces a minimal block decomposition (one with the fewest blocks), but since any such block decomposition must be incompressible, these algorithms are a  ... 
doi:10.1016/j.jpaa.2014.07.030 fatcat:rqhibi2tjbajlhm5bvpyrz3yb4

Numerical primary decomposition [article]

Anton Leykin
2008 arXiv   pre-print
We propose an algorithm to produce a collection of witness sets that contains a NPD and that can be used to solve the ideal membership problem for I.  ...  Consider an ideal I ⊂ R = [x_1,...,x_n] defining a complex affine variety X ⊂^n. We describe the components associated to I by means of numerical primary decomposition (NPD).  ...  Another example is a solution to the ideal membership problem given via NPD.  ... 
arXiv:0801.3105v2 fatcat:jgqa6qojljexzoszjrdslekw2a

The Canonical Decomposition of $\mathcal{C}^n_d$ and Numerical Gröbner and Border Bases

Kim Batselier, Philippe Dreesen, Bart De Moor
2014 SIAM Journal on Matrix Analysis and Applications  
An SVD-based algorithm is presented that numerically computes the canonical decomposition.  ...  The SVD-based canonical decomposition algorithm is also extended to numerically compute border bases. A tolerance for each of the algorithms is derived using perturbation theory of principal angles.  ...  All algorithms are illustrated with numerical examples. To our knowledge, no SVD-based method to compute a Gröbner basis has been proposed yet.  ... 
doi:10.1137/130927176 fatcat:6xmh7zjv3bdvtd57al6ody27dy

Noether normalization guided by monomial cone decompositions

Daniel Robertz
2009 Journal of symbolic computation  
Such a decomposition of the complement of the corresponding initial ideal in the set of all monomials -also known as a Stanley decomposition -is constructed in the context of Janet bases, in order to come  ...  up with sparse coordinate changes which achieve Noether normal position for the given ideal.  ...  We call the cone decomposition of S (of Mon(R) \ S) which is constructed by Algorithm 1 (resp. 2) the Janet decomposition of S (resp. of Mon(R) \ S, or of R/I if S = lm(I) for an ideal I of R).  ... 
doi:10.1016/j.jsc.2009.02.004 fatcat:7zadhdynzjcahikkak5bvljwim
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