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Irreducible decomposition of polynomial ideals

2005
*
Journal of symbolic computation
*

In this paper we present some algorithms for computing an

doi:10.1016/j.jsc.2004.11.005
fatcat:xawnmuw35bcsxjbngqsjb25gha
*irreducible**decomposition**of*an*ideal*in a*polynomial*ring R = K [x 1 , . . . , x n ] where K is an arbitrary effective field. ... Introduction In this paper we present some algorithms for computing an*irreducible**decomposition**of*an*ideal*in a*polynomial*ring R = K [x 1 , . . . , x n ] where K is an arbitrary effective field. ... Some*of*the earliest proofs*of*primary*decomposition*were based on the existence*of*an*irreducible**decomposition*, using the fact that every*irreducible**ideal*is primary, even though not every primary*ideal*...##
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Prime Decompositions of Radicals in Polynomial Rings

1994
*
Journal of symbolic computation
*

We show that prime

doi:10.1006/jsco.1994.1052
fatcat:36lszgijy5hnhib56ckk4swolq
*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 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. ...##
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Comparison of probabilistic algorithms for analyzing the components of an affine algebraic variety

2014
*
Applied Mathematics and Computation
*

Given a system

doi:10.1016/j.amc.2013.12.165
fatcat:3loemn6l6ba6phdqv3s64nu6uu
*of**polynomials*f (z), the numerical*irreducible**decomposition**of*V (f ) consists*of*a witness set W i,j for each*irreducible*component Z i,j . ... Or, Z is*irreducible*if and only if I(Z) is a prime*ideal*; furthermore, if Z is reducible, its break-up into*irreducible*components corresponds to a*decomposition**of*I(Z) as an intersection*of*prime*ideals*...##
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A Numerical Local Dimension Test for Points on the Solution Set of a System of Polynomial Equations

2009
*
SIAM Journal on Numerical Analysis
*

For example, one may compute the isolated solutions

doi:10.1137/08073264x
fatcat:5sd5fjlcvrehflsgrwl72nhcpy
*of*a*polynomial*system without having to carry out the full numerical*irreducible**decomposition*. ... computation*of*a numerical*irreducible**decomposition*. ... It is important to note that the numerical*irreducible**decomposition*corresponds to a prime*decomposition**of*the radical*of*an*ideal*rather than a primary*decomposition*. ...##
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The Schonemann-Eisenstein Irreducibility Criteria in Terms of Prime Ideals

1938
*
Transactions of the American Mathematical Society
*

Hence the generalizations

doi:10.2307/1990040
fatcat:ptr6mz7hhbgg3pqj27wp3ag7mm
*of*the Eisenstein criterion are merely statements about prime*ideal**decompositions*. 6.*Irreducibility**of**polynomials*in several variables. ... In ¡0, every*ideal*B which is not a divisor*of*zero* has a*decomposition*, unique except for the order*of*factors, as a product*of*prime*ideals*from £). ...##
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The Schönemann-Eisenstein irreducibility criteria in terms of prime ideals

1938
*
Transactions of the American Mathematical Society
*

Hence the generalizations

doi:10.1090/s0002-9947-1938-1501940-x
fatcat:tj7aa422njcd5f7ioplijiyxsa
*of*the Eisenstein criterion are merely statements about prime*ideal**decompositions*. 6.*Irreducibility**of**polynomials*in several variables. ... In ¡0, every*ideal*B which is not a divisor*of*zero* has a*decomposition*, unique except for the order*of*factors, as a product*of*prime*ideals*from £). ...##
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Page 2 of Annals of Mathematics Vol. 37, Issue 1
[page]

1936
*
Annals of Mathematics
*

any

*decomposition*field or considering any*decomposition**of*(x, a) into*irreducible*factors in Fla] before setting up the algorithm. ... In the present paper we obtain, very simply, the*decomposition*field*of*f(x) by first giving a finite algorithm for obtaining the*irreducible*factors*of*o(x, a) in Fla] without assuming the existence*of*...##
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Computing irreducible representations of finite groups

1990
*
Mathematics of Computation
*

In particular, it follows that some representative

doi:10.1090/s0025-5718-1990-1035925-1
fatcat:theuji7o7bd77fribcpi34apqe
*of*each equivalence class*of**irreducible*representations admits a*polynomial*-size description. ... We present a*polynomial*-time algorithm to find a complete set*of*nonequivalent*irreducible*representations over the field*of*complex numbers*of*a finite group given by its multiplication table. ... Thus, from an*irreducible*F-module we can efficiently find a*decomposition**of*B into a direct sum*of*minimal left*ideals*. ...##
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Computing Irreducible Representations of Finite Groups

1990
*
Mathematics of Computation
*

In particular, it follows that some representative

doi:10.2307/2008443
fatcat:fqjtiyjmu5cgjkkwdbt3ha7pom
*of*each equivalence class*of**irreducible*representations admits a*polynomial*-size description. ... We present a*polynomial*-time algorithm to find a complete set*of*nonequivalent*irreducible*representations over the field*of*complex numbers*of*a finite group given by its multiplication table. ... Thus, from an*irreducible*F-module we can efficiently find a*decomposition**of*B into a direct sum*of*minimal left*ideals*. ...##
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Ideal Theory in Rings (Translation of "Idealtheorie in Ringbereichen" by Emmy Noether)
[article]

2014
*
arXiv
*
pre-print

*of*coprime

*irreducible*

*ideals*; equivalent concepts regarding modules. ...

*of*prime

*ideals*with primary

*ideals*; the representation

*of*an

*ideal*as the least common multiple

*of*relatively prime

*irreducible*

*ideals*; isolated

*ideals*; the representation

*of*an

*ideal*as the product ... The

*decomposition*into coprime

*irreducible*

*ideals*is given by Schmeidler 4 for the

*polynomial*ring, using elimination theory for the proof

*of*finiteness. ...

##
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Decomposition of primes in number fields defined by trinomials

1991
*
Séminaire de Théorie des Nombres de Bordeaux
*

An integer

doi:10.5802/jtnb.40
fatcat:jz2zbc6wevgsdfmt4n5sriezjq
*ideal*a*of*any number field L will be called "q analogous to the*polynomial*F(X)" if the*decomposition**of*a into a product*of*prime*ideals**of*L is*of*the type : 2.1. ... The*decomposition**of*q into a product*of*prime*ideals**of*li is a ~follows : If vq(B) > vq(A) and q la, If q JAB and the*decomposition**of*f (X ) into a product*of**irreducible*factors (mod q) is*of*...##
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Essential Components of an Algebraic Differential Equation

1999
*
Journal of symbolic computation
*

We present an algorithm to determine the essential singular components

doi:10.1006/jsco.1999.0319
fatcat:kowqm75ilfflvc3x75wyrqnqxq
*of*an algebraic differential equation. ... I appreciated the work and the comments*of*the referees. I would also like to thank G. Labahn and M. Singer for their sensible comments in the writing*of*this paper. ...*of*an*irreducible*differential*polynomial*. ...##
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Page 5859 of Mathematical Reviews Vol. , Issue 96j
[page]

1996
*
Mathematical Reviews
*

In order to prove these results the authors undertake the study

*of*the*irreducible*elements*of*Int(D), especially among the*irreducible**polynomials**of*K[X]. ... They also provide examples*of**polynomials*that are*irreducible*in Int(D) but not in K[X]. ...##
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Irreducible Decomposition of Curves

2002
*
Journal of symbolic computation
*

In this paper, we propose a fast semi-numerical algorithm for computing all

doi:10.1006/jsco.2000.0528
fatcat:uvn5r6tfezamngmtd56744hkx4
*irreducible*branches*of*a curve in C τ defined by*polynomials*with rational coefficients, we also treat the case*of*a non-reduced ... Our approach could be applied to more general situations, it generalizes our previous study on absolute factorization*of**polynomials*. ...*irreducible*components A space curve C will be represented by a set*of*generators*of*an*ideal*I spanned by*polynomials*with coefficients in Q. ...##
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Characteristic set method for differential–difference polynomial systems

2009
*
Journal of symbolic computation
*

CS Method
DD-

doi:10.1016/j.jsc.2008.02.010
fatcat:53uze3dujja2fmjv336to5ee4m
*Polynomials*DD-Chains Zero*Decomposition**Irreducible*Chain A regular chain A is*irreducible*: A i is*irreducible*in y i module A 1 = 0, . . . , A i−1 = 0 Example. ... For a regular chain, the following properties are equivalent 1 A is*irreducible*; 2 asat(A) is a prime*ideal**of*dimension dim(A). ... Parameters*of*Lemma. P invertible wrt A ⇒ δP invertible wrt A. Lemma. Proper*irreducibility*implies DD-kernels. ...
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