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

Yongbin Li
2016 arXiv   pre-print
In this note, we show that 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

Deepak Kapur
2006 Journal of symbolic computation
This is done 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

Masayuki Noro, Kazuhiro Yokoyama
2004 Journal of symbolic computation
An algorithm for 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 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?

Daniel Lazard
2009 Journal of symbolic computation
Then I describe 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

Diego Cifuentes, Pablo A. Parrilo
2017 SIAM Journal on applied algebra and geometry
Chordal networks can be 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

Xin Gao
2012 Journal of Applied & Computational Mathematics
Because there are 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.  ...

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

Victor Magron and Mohab Safey El Din and Trung-Hieu Vu
2021 arXiv   pre-print
This is usually tackled through 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 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

Evelyne Hubert
2000 Journal of symbolic computation
We present original results in constructive algebra that makes 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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