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Irreducible decomposition of polynomial ideals

E. Fortuna, P. Gianni, B. Trager
2005 Journal of symbolic computation  
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.  ...  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  ... 
doi:10.1016/j.jsc.2004.11.005 fatcat:xawnmuw35bcsxjbngqsjb25gha

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 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.  ... 
doi:10.1006/jsco.1994.1052 fatcat:36lszgijy5hnhib56ckk4swolq

Comparison of probabilistic algorithms for analyzing the components of an affine algebraic variety

Daniel J. Bates, Wolfram Decker, Jonathan D. Hauenstein, Chris Peterson, Gerhard Pfister, Frank-Olaf Schreyer, Andrew J. Sommese, Charles W. Wampler
2014 Applied Mathematics and Computation  
Given a system 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  ... 
doi:10.1016/j.amc.2013.12.165 fatcat:3loemn6l6ba6phdqv3s64nu6uu

A Numerical Local Dimension Test for Points on the Solution Set of a System of Polynomial Equations

Daniel J. Bates, Jonathan D. Hauenstein, Chris Peterson, Andrew J. Sommese
2009 SIAM Journal on Numerical Analysis  
For example, one may compute the isolated solutions 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.  ... 
doi:10.1137/08073264x fatcat:5sd5fjlcvrehflsgrwl72nhcpy

The Schonemann-Eisenstein Irreducibility Criteria in Terms of Prime Ideals

Saunders MacLane
1938 Transactions of the American Mathematical Society  
Hence the generalizations 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 £).  ... 
doi:10.2307/1990040 fatcat:ptr6mz7hhbgg3pqj27wp3ag7mm

The Schönemann-Eisenstein irreducibility criteria in terms of prime ideals

Saunders MacLane
1938 Transactions of the American Mathematical Society  
Hence the generalizations 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 £).  ... 
doi:10.1090/s0002-9947-1938-1501940-x fatcat:tj7aa422njcd5f7ioplijiyxsa

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  ... 

Computing irreducible representations of finite groups

L{ászl{ó Babai, Lajos R{ónyai
1990 Mathematics of Computation  
In particular, it follows that some representative 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.  ... 
doi:10.1090/s0025-5718-1990-1035925-1 fatcat:theuji7o7bd77fribcpi34apqe

Computing Irreducible Representations of Finite Groups

Laszlo Babai, Lajos Ronyai
1990 Mathematics of Computation  
In particular, it follows that some representative 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.  ... 
doi:10.2307/2008443 fatcat:fqjtiyjmu5cgjkkwdbt3ha7pom

Ideal Theory in Rings (Translation of "Idealtheorie in Ringbereichen" by Emmy Noether) [article]

Daniel Berlyne
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.  ... 
arXiv:1401.2577v1 fatcat:4cbgcpyhgbd45gd4wmzyf4uxmu

Decomposition of primes in number fields defined by trinomials

P. Llorente, E. Nart, N. Vila
1991 Séminaire de Théorie des Nombres de Bordeaux  
An integer 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  ... 
doi:10.5802/jtnb.40 fatcat:jz2zbc6wevgsdfmt4n5sriezjq

Essential Components of an Algebraic Differential Equation

Evelyne Hubert
1999 Journal of symbolic computation  
We present an algorithm to determine the essential singular components 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.  ... 
doi:10.1006/jsco.1999.0319 fatcat:kowqm75ilfflvc3x75wyrqnqxq

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].  ... 

Irreducible Decomposition of Curves

André Galligo, David Rupprecht
2002 Journal of symbolic computation  
In this paper, we propose a fast semi-numerical algorithm for computing all 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.  ... 
doi:10.1006/jsco.2000.0528 fatcat:uvn5r6tfezamngmtd56744hkx4

Characteristic set method for differential–difference polynomial systems

X.S. Gao, J. Van der Hoeven, C.M. Yuan, G.L. Zhang
2009 Journal of symbolic computation  
CS Method DD-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.  ... 
doi:10.1016/j.jsc.2008.02.010 fatcat:53uze3dujja2fmjv336to5ee4m
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