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Computing the decomposition group of a zero-dimensional ideal by elimination method
[article]

2016
*
arXiv
*
pre-print

In this note, we show that

arXiv:1601.06626v1
fatcat:6phakvrdlfbgnopuv34lh4ncxu
*the**decomposition**group*Dec(I)*of**a**zero*-*dimensional*radical*ideal*I in K[x_1,... ...*The*new*method*makes*a*theoretical contribution to discuss*the**decomposition**group**of*I*by*using*Computer*Algebra without considering*the*complexity. ... An alternative*method*: Given*a**zero*-*dimensional*radical*ideal*I in K[x], we try to*compute*Dec(I) as follows. Step 1.*Compute**a*Gröbner basis*of*I. Step 2.*Compute*each m x i for 1 ≤ i ≤ n. Step 3. ...##
###
Preface on the contributed papers

2006
*
Journal of symbolic computation
*

This is done

doi:10.1016/j.jsc.2005.09.006
fatcat:763hbyncf5gs7fznrds64vcyye
*by*first showing how to*compute**a*basis*of**the*inverse system*of**a*given*zero*-*dimensional**ideal*. ... Preface Preface on*the*contributed papers*The*paper*by*Heis et al. generalizes*the*well-known notion*of*square-free*decomposition**of**a*univariate polynomial to*zero*-*dimensional**ideals*over*a*multivariate ...*a*specific set*of*cut-reduction rules yield*a*resolution proof which is subsumed*by**a*resolution proof*of**the*characteristic clause set. ...##
###
Page 2617 of Mathematical Reviews Vol. , Issue 95e
[page]

1995
*
Mathematical Reviews
*

Finally,

*a**method*to isolate and approximate*the*real*zeroes**of**a**zero*-*dimensional**ideal*over*the*rationals*by*means*of*Sturm sequences is given. ... Radicals and primary*decompositions*in higher dimensions are treated*by*extension to*a**zero*-*dimensional*situation. ...##
###
Implementation of prime decomposition of polynomial ideals over small finite fields

2004
*
Journal of symbolic computation
*

An algorithm for

doi:10.1016/j.jsc.2003.08.004
fatcat:xskdukgarrdpjeuqz3eksrfhwq
*the*prime*decomposition**of*polynomial*ideals*over small finite fields is proposed and implemented on*the*basis*of*previous work*of**the*second author. ...*The*practicality*of**the*algorithm is examined*by*testing*the*implementation experimentally, which also reveals information about*the*quality*of**the*implementation. ... Incremental intermediate*decomposition*Intermediate*decomposition**of**the*radical*of**a**zero*-*dimensional**ideal*J can be performed*by*extracting non-trivial*ideals*from*the*set*of**ideals*: {I d K [X ] (J, ...##
###
Index to Volume 41

2006
*
Journal of symbolic computation
*

devoted to symbolic

doi:10.1016/s0747-7171(06)00119-2
fatcat:n3voc3w2dvex7bo3gopguifv4m
*computation*, 255 Bruno Buchberger's PhD thesis 1965: An algorithm for finding*the*basis elements*of**the*residue class ring*of**a**zero**dimensional*polynomial*ideal*, 475 BUCHBERGER, B ... ., Bruno Buchberger's PhD thesis 1965: An algorithm for finding*the*basis elements*of**the*residue class ring*of**a**zero**dimensional*polynomial*ideal*, 475 BUCHBERGER, B., Comments on*the*translation*of*my ... ., On inverse systems and squarefree*decomposition**of**zero*-*dimensional*polynomial*ideals*, 261 HENDERSON, R., MACEDO, T. and NELSON, S., Symbolic*computation*with finite quandles, 811 HEUBERGER, C., All ...##
###
Contents and Index of Volume 34

2002
*
Journal of symbolic computation
*

*Zero*-

*dimensional*

*Ideals*, 451 Mosteig, E. and Sweedler, M., Valuations and Filtrations, 399 MultInt,

*a*MAPLE Package for Multiple Integration

*by*

*the*WZ

*Method*, 329 On

*the*Resolution

*of*Resultant Type Equations ...

*the*Primary

*Decomposition*

*of*

*Zero*-

*dimensional*

*Ideals*. . . . . . . . . . . . . . . . 451 Xia, B. and Yang, L., An Algorithm for Isolating

*the*Real Solutions

*of*Semi-algebraic Systems ...

##
###
Thirty years of Polynomial System Solving, and now?

2009
*
Journal of symbolic computation
*

Then I describe

doi:10.1016/j.jsc.2008.03.004
fatcat:umelra7aarag3fcsehfu67is6u
*the*main challenges which are now opened*by**the*availability*of*efficient*zero*-*dimensional*solvers. ... In this introductory paper to*the*special issue, I describe first my personal view*of**the*history*of*Polynomial System Solving during my career. ... Dimension*of**the*(complex) variety*of**the**zeros**of*an*ideal*. This is easily deduced from*the*Gröbner basis*of**the**ideal*, for any monomial ordering. Real*zero*-*dimensional*solving. ...##
###
Chordal Networks of Polynomial Ideals

2017
*
SIAM Journal on applied algebra and geometry
*

Chordal networks can be

doi:10.1137/16m106995x
fatcat:sj3qlirgzngplhobzzudlogusa
*computed*for arbitrary polynomial systems using*a*refinement*of**the*chordal*elimination*algorithm from [Cifuentes-Parrilo-2016]. ...*The*sparsity structure*of**a*polynomial system is often described*by**a*graph that captures*the*interactions among*the*variables. ... We say that F is chordally*zero*-*dimensional*if for each maximal clique X l*of*graph G*the**ideal*F ∩ K[X l ] is*zero*-*dimensional*. ...##
###
Page 3939 of Mathematical Reviews Vol. , Issue 2000f
[page]

2000
*
Mathematical Reviews
*

*of*

*zero*-

*dimensional*

*ideals*. ... Summary: “To give an efficiently

*computable*representation

*of*

*the*

*zeros*

*of*

*a*

*zero*-

*dimensional*

*ideal*J, F. Rouillier [*Résolution des systemes

*zéro*-dimensionnels”, Ph.D. Thesis, Univ. ...

##
###
Mathematical Approaches to the NMR Peak-Picking Problem

2012
*
Journal of Applied & Computational Mathematics
*

Because there are

doi:10.4172/2168-9679.1000e103
fatcat:xvvdzvvjdfbivlo57zlboi7yue
*a*finite number*of*atoms in*a*protein,*the*number*of*peaks is also finite.*Ideally*,*the*non-signal regions*of**the*spectrum should have intensity values equal to*zero*. ... Among all*the*signals,*the*strong and obvious peaks are easy to identify*by**computational**methods*. ...##
###
Book reports

2005
*
Computers and Mathematics with Applications
*

*The*GTZ Scheme. 35.4. Higher-

*Dimensional*

*Decomposition*Algorithms. 35.5.

*Decomposition*Algorithms for Allgemeine

*Ideals*. 35.5.1.

*Zero*-

*Dimensional*Allgemeine

*Ideals*. 35.5.2. ... Squarefree

*Decomposition*

*of*

*a*

*Zero*-

*dimensional*

*Ideal*. 36. Macaulay III. 36.1. Hilbert Function and Complete Intersections. 36.2.

*The*Coefficients

*of*

*the*Hilbert Function. 36.3. Perfectness. 37. ...

##
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Sum of Squares Decompositions of Polynomials over their Gradient Ideals with Rational Coefficients
[article]

2021
*
arXiv
*
pre-print

This is usually tackled through

arXiv:2107.11825v1
fatcat:v2pt6fhzmbgndiwelo3qvyao7m
*the**computation**of*sums-*of*-squares*decompositions*which rely on efficient numerical solvers for semi-definite programming. This*method*faces two difficulties. ... Assessing non-negativity*of*multivariate polynomials over*the*reals, through*the**computation**of*certificates*of*non-negativity, is*a*topical issue in polynomial optimization. ... Precisely, assuming*the*gradient*ideal*I grad (f ) generated*by*all partial derivatives*of*f is*zero*-*dimensional*and radical, and that f reaches its minimum over R n , this algorithm*computes*an SOS*decomposition*...##
###
Contents of Volume 41

2006
*
Journal of symbolic computation
*

PhD thesis . . . . . . . 471 Buchberger, B., Bruno Buchberger's PhD thesis 1965: An algorithm for finding

doi:10.1016/s0747-7171(06)00118-0
fatcat:r3qfch5zsncrhed5wn3uex37ce
*the*basis elements*of**the*residue class ring*of**a**zero**dimensional*polynomial*ideal*. . 475 ... restrictions*of**ideals*in finitely generated k-algebras*by*means*of*Buchberger's algorithm . . . . . 372 Baaz, M. and Leitsch,*A*., Towards*a*clausal analysis*of*cut-*elimination*. . . . . 381 Nakagawa ...##
###
Page 113 of Mathematical Reviews Vol. , Issue 98A
[page]

1998
*
Mathematical Reviews
*

Symbolic

*Comput*. 22 (1996), no. 3, 247-277.*A*new algorithm is given to*compute**the*primary decomposi- tion*of**a*(not necessarily*zero*-*dimensional*) polynomial*ideal*. ...*The*first goal is achieved*by**computing**the*prime*decomposition**of**the*radical,*decomposition**of**the**ideal*into pseudo-primary*ideals*(i.e.,*ideals*whose radical is prime), and extraction*of**the*prime ...##
###
Factorization-free Decomposition Algorithms in Differential Algebra

2000
*
Journal of symbolic computation
*

We present original results in constructive algebra that makes

doi:10.1006/jsco.1999.0344
fatcat:bhwx5bq6lfdptkzmddnuxgshpi
*the*algorithm exible and simple. ... We present an e ective version*of*Ritt's algorithm. We apply material*of*(Boulier et al. 1995) for which we give new concise proofs. ... We can thus replace*a*j*by*its primitive part according to x j . Similarly, to decrease*the*degrees, we can*eliminate*from*a*j*the*factors which are common with s j . ...
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