## A Gröbner free alternative for polynomial system solving (2001)

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Venue: | Journal of Complexity |

Citations: | 82 - 17 self |

### BibTeX

@ARTICLE{Giusti01agröbner,

author = {Marc Giusti and Grégoire Lecerf and Bruno Salvy},

title = {A Gröbner free alternative for polynomial system solving},

journal = {Journal of Complexity},

year = {2001},

volume = {17},

pages = {154--211}

}

### Years of Citing Articles

### OpenURL

### Abstract

Given a system of polynomial equations and inequations with coefficients in the field of rational numbers, we show how to compute a geometric resolution of the set of common roots of the system over the field of complex numbers. A geometric resolution consists of a primitive element of the algebraic extension defined by the set of roots, its minimal polynomial and the parametrizations of the coordinates. Such a representation of the solutions has a long history which goes back to Leopold Kronecker and has been revisited many times in computer algebra. We introduce a new generation of probabilistic algorithms where all the computations use only univariate or bivariate polynomials. We give a new codification of the set of solutions of a positive dimensional algebraic variety relying on a new global version of Newton’s iterator. Roughly speaking the complexity of our algorithm is polynomial in some kind of degree of the system, in its height, and linear in the complexity of evaluation

### Citations

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Citation Context ...f degree n in terms of number of operations in the base ring. Many authors have contributed to these topics. Some very good historical presentations can be found in the books of Aho, Hopcroft, Ullman =-=[4]-=-, Bürgisser, Clausen, Shokrolahi [14], Bini, Pan [10] among others. Let R be a unitary commutative ring, the Schönhage-Strassen polynomial multiplication [71, 70, 63] of two polynomials of R[T ] of de... |

803 |
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Citation Context ...ommon divisor has complexity M(n) [69]. Let R be a unitary ring, the multiplication of two n × n matrices can be done in O(n ω ) arithmetic operations in R. The exponent ω can be taken less than 2.39 =-=[21]-=-. If R is a field, Bunch and Hopcroft showed that matrix inversion is not harder than the multiplication [13]. According to [13], the converse fact is due to Winograd. In our case, R is a k-algebra k[... |

622 |
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Citation Context ...aim of this article is to demonstrate that our algorithm has a practical interest and is competitive with the other methods. We have implemented our algorithm within the computer algebra system Magma =-=[1, 16, 12]-=-, the package has been called Kronecker [55] and is available with its documentation at http://www.gage.polytechnique.fr/~lecerf/software/kronecker/. We compare our implementation to Gröbner bases com... |

401 | Deforestation: Transforming programs to eliminate trees
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Citation Context ...a new estimate of the exponents of the complexity of Theorem [37] above improving the results of [45]. One main step of this transformation is obtained by a technique reminiscent of the deforestation =-=[83]-=-, that we had already used in [39] to replace straightline programs by an efficient use of specialization. We only need polynomials in at most two variables. From a geometrical point of view our algor... |

398 | Fast probabilistic algorithms for verification of polynomial identities
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Citation Context ...a lucky p is out of the scope of this work but such considerations are as in [40, 41, 42]. The probability of failure of the algorithm given in [45] has been studied using Zippel-Schwartz’s zero test =-=[85, 72]-=- for multivariate polynomials. We could use the same analysis here to quantify the probability mentioned in Theorem 1, but this has no practical interest without the quantification of the probability ... |

336 |
Gröbner bases, a computational Approach to Commutative Algebra
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Citation Context ...ar [65], Giusti, Heintz [34], Alonso, Becker, Roy, Wörman [6] and many others. From a practical point of view, the computation of such a representation is always relying on Gröbner basis computations =-=[84]-=-, either with a pure lexicographical elimination order, or with an algorithm of change of basis [28, 20] or from any basis using a generalization of Newton’s formulæ by Rouillier [66, 67]. In 1995, Gi... |

292 |
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Citation Context ...tic since the number of solutions found is equal to the Bézout number of the system. 8.3 Camera Calibration (Kruppa) The original problem comes from [52] and has been introduced in computer vision in =-=[60]-=-. It is composed of 5 equations in 5 variables. Each equation is a difference of two products of two linear forms. The parameter h is the size of the integers of the input system. The systems have 32 ... |

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240 |
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182 |
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Citation Context ...rations in the base ring. Many authors have contributed to these topics. Some very good historical presentations can be found in the books of Aho, Hopcroft, Ullman [4], Bürgisser, Clausen, Shokrolahi =-=[14]-=-, Bini, Pan [10] among others. Let R be a unitary commutative ring, the Schönhage-Strassen polynomial multiplication [71, 70, 63] of two polynomials of R[T ] of degree at most n can be performed in O(... |

173 |
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Citation Context ...y authors, in the particular zero-dimensional case: Chistov, Grigoriev [19], Canny [17], Gianni, Mora [32], Kobayashi, Fujise, Furukawa [47], Heintz, Roy, Solerno [46], Lakshman, Lazard [54], Renegar =-=[65]-=-, Giusti, Heintz [34], Alonso, Becker, Roy, Wörman [6] and many others. From a practical point of view, the computation of such a representation is always relying on Gröbner basis computations [84], e... |

156 |
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Citation Context ...t of view, the computation of such a representation is always relying on Gröbner basis computations [84], either with a pure lexicographical elimination order, or with an algorithm of change of basis =-=[28, 20]-=- or from any basis using a generalization of Newton’s formulæ by Rouillier [66, 67]. In 1995, Giusti, Heintz, Morais and Pardo [36, 64] rediscovered Kronecker’s approach without any prior knowledge of... |

142 |
The Algebraic Theory of Modular Systems
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125 |
The complexity of partial derivatives
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Citation Context ...ations and assumptions, the complexity of Algorithm 1 returning a solution of f1, . . . , fn at precision Iκ (where κ is a power of 2) is in O � (nL + n Ω log2 (κ) � )M(δ) a(2 j ) � . Proof Thanks to =-=[8]-=-, we only need at most 5L operations to evaluate the gradient of a straight-line program of size L. Thus the evaluation of the polynomials f and the Jacobian matrix J of Algorithm 1 has complexity O(n... |

115 |
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Citation Context ...ementary operations on n × n matrices over any commutative ring R in terms of arithmetic operations in R: addition, multiplication, determinant and adjoint matrix. In fact, Ω can be taken less than 4 =-=[3, 9, 22, 56]-=-, see also [79, 59]. 4 Global Newton Lifting In this section we present the new global Newton-Hensel iterator. First, through an example, we recall the Newton-Hensel method in its local form and show ... |

114 | Introduction to Commutative Algebra and Algebraic Geometry - Kunz - 1985 |

101 |
The Computational Complexity of Algebraic and Numeric Problems
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Citation Context ...0, 63] of two polynomials of R[T ] of degree at most n can be performed in O(n log(n) log log(n)) arithmetic operations in R. The division of polynomials has the same complexity as the multiplication =-=[11, 78]-=-. The greatest common divisor of two polynomials of degree at most n over a field K can be computed in M(n) arithmetic operations in K [61]. The resultant, the sub-resultants and the interpolation can... |

97 |
Solving zero-dimensional systems through the rational univariate representation
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Citation Context ...sis computations [84], either with a pure lexicographical elimination order, or with an algorithm of change of basis [28, 20] or from any basis using a generalization of Newton’s formulæ by Rouillier =-=[66, 67]-=-. In 1995, Giusti, Heintz, Morais and Pardo [36, 64] rediscovered Kronecker’s approach without any prior knowledge of it and improved the space complexity but not the running time complexity. A first ... |

88 |
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Citation Context ...ementary operations on n × n matrices over any commutative ring R in terms of arithmetic operations in R: addition, multiplication, determinant and adjoint matrix. In fact, Ω can be taken less than 4 =-=[3, 9, 22, 56]-=-, see also [79, 59]. 4 Global Newton Lifting In this section we present the new global Newton-Hensel iterator. First, through an example, we recall the Newton-Hensel method in its local form and show ... |

78 |
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Citation Context ...ound in the books of Aho, Hopcroft, Ullman [4], Bürgisser, Clausen, Shokrolahi [14], Bini, Pan [10] among others. Let R be a unitary commutative ring, the Schönhage-Strassen polynomial multiplication =-=[71, 70, 63]-=- of two polynomials of R[T ] of degree at most n can be performed in O(n log(n) log log(n)) arithmetic operations in R. The division of polynomials has the same complexity as the multiplication [11, 7... |

71 |
Definability and fast quantifier elimination in algebraically closed fields, Theor
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Citation Context ...inear combination of the equations to recover this situation, as showed in [49, Proposition 37]. The number δ is defined as max(deg(V1), . . . , deg(Vn−1)), it is bounded by d n , by Bézout’s theorem =-=[43]-=- (see also [81] and [30]). A precise definition of geometric resolutions is given in §3.2. The geometric resolution returned by the algorithm underlying the above theorem has its integers represented ... |

66 |
Grundzüge einer arithmetischen Theorie der algebraischen Grössen
- Kronecker
(Show Context)
Citation Context ...tation of a variety in the form of (1) above has a long history. To the best of our knowledge the oldest trace of this representation is to be found in Kronecker’s work at the end of the 19th century =-=[51]-=- and a few years later in König’s [48]. Their representation is naturally defined for positive dimensional algebraic varieties, for instance for a variety of codimension i it has the form: q(x1, . . .... |

63 |
Exact solution of linear equations using p-adic expansions
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Citation Context ...In this case there exists a unique rational geometric resolution lying over the modular one; the lifting process gives the p-adic expansions of its rational coefficients at any required precision. In =-=[24]-=- Dixon gave a Padé approximant method for integers, see also [41] and [40] for related results. Proposition 7 [24] Let s, h > 1 be integers and suppose that there exist integers f, g such that gs ≡ f(... |

62 | Lower bounds for diophantine approximations
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(Show Context)
Citation Context ...ut any prior knowledge of it and improved the space complexity but not the running time complexity. A first breakthrough was obtained by Giusti, Hägele, Heintz, Montaña, Morais, Morgenstern and Pardo =-=[33, 35]-=-: there exists an algorithm with a complexity roughly speaking polynomial in the degree of the system, in its height and in the number of the variables. Then, in [37], it is announced that the height ... |

61 | When polynomial equation systems can be “solved” fast
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(Show Context)
Citation Context ...hical elimination order, or with an algorithm of change of basis [28, 20] or from any basis using a generalization of Newton’s formulæ by Rouillier [66, 67]. In 1995, Giusti, Heintz, Morais and Pardo =-=[36, 64]-=- rediscovered Kronecker’s approach without any prior knowledge of it and improved the space complexity but not the running time complexity. A first breakthrough was obtained by Giusti, Hägele, Heintz,... |

60 |
Schnelle Berechnung von Kettenbruchenwicklungen
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Citation Context ...ying two integers of bitsize at most n has a bit-complexity in O(n log(n) log log(n)). The division has the same complexity as the multiplication [73]. The greatest common divisor has complexity M(n) =-=[69]-=-. Let R be a unitary ring, the multiplication of two n × n matrices can be done in O(n ω ) arithmetic operations in R. The exponent ω can be taken less than 2.39 [21]. If R is a field, Bunch and Hopcr... |

58 |
The fundamental theorem of algebra and complexity theory
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Citation Context ... numerical algorithms is the certification of the result. In [18] a comparison is made between our method and the numerical approach using homotopy and the approximate zero theory introduced by Smale =-=[74, 75]-=-. 2.2 Description of the Computations We present now the actual computations performed by our algorithm in a generic case. Our algorithm is incremental in the number of equations to be solved. Let Si ... |

57 | Straight–line programs in geometric elimination theory
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- 1998
(Show Context)
Citation Context ...ut any prior knowledge of it and improved the space complexity but not the running time complexity. A first breakthrough was obtained by Giusti, Hägele, Heintz, Montaña, Morais, Morgenstern and Pardo =-=[33, 35]-=-: there exists an algorithm with a complexity roughly speaking polynomial in the degree of the system, in its height and in the number of the variables. Then, in [37], it is announced that the height ... |

57 |
A computational method for Diophantine approximation
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(Show Context)
Citation Context ... in §2.2.2 and §6. This improves [45, §4.2] by avoiding the use of primitive elements computations in two variables which is used twice in [62]. This technique first appeared in [34] and developed in =-=[50]-=-. The last step is the intensive use of modular arithmetic: the resolution is computed modulo a small prime number, the integers are lifted at the end by our global Newton iterator. Hence the cost of ... |

56 |
Schnelle Multiplikation von Polynomen über Körpern der Charakteristik 2
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- 1977
(Show Context)
Citation Context ...ound in the books of Aho, Hopcroft, Ullman [4], Bürgisser, Clausen, Shokrolahi [14], Bini, Pan [10] among others. Let R be a unitary commutative ring, the Schönhage-Strassen polynomial multiplication =-=[71, 70, 63]-=- of two polynomials of R[T ] of degree at most n can be performed in O(n log(n) log log(n)) arithmetic operations in R. The division of polynomials has the same complexity as the multiplication [11, 7... |

53 | Zeroes, multiplicities and idempotents for zerodimensional systems
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Citation Context ...stov, Grigoriev [19], Canny [17], Gianni, Mora [32], Kobayashi, Fujise, Furukawa [47], Heintz, Roy, Solerno [46], Lakshman, Lazard [54], Renegar [65], Giusti, Heintz [34], Alonso, Becker, Roy, Wörman =-=[6]-=- and many others. From a practical point of view, the computation of such a representation is always relying on Gröbner basis computations [84], either with a pure lexicographical elimination order, o... |

52 |
Triangular factorization and inversion by fast matrix multiplication
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(Show Context)
Citation Context ... be done in O(n ω ) arithmetic operations in R. The exponent ω can be taken less than 2.39 [21]. If R is a field, Bunch and Hopcroft showed that matrix inversion is not harder than the multiplication =-=[13]-=-. According to [13], the converse fact is due to Winograd. In our case, R is a k-algebra k[T ]/q(T ), where q is a square-free monic polynomial of k[T ], so we can not apply the results of [13] to com... |

49 | Algebraic solution of systems of polynomial equations using Gröbner bases. In Applied algebra, algebraic algorithms and error-correcting codes (Menorca
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Citation Context ...his representation has been used in computer algebra as a tool to obtain complexity results by many authors, in the particular zero-dimensional case: Chistov, Grigoriev [19], Canny [17], Gianni, Mora =-=[32]-=-, Kobayashi, Fujise, Furukawa [47], Heintz, Roy, Solerno [46], Lakshman, Lazard [54], Renegar [65], Giusti, Heintz [34], Alonso, Becker, Roy, Wörman [6] and many others. From a practical point of view... |

49 | La détermination des points isolés et de la dimension d'une variété algebrique peut se faire en temps polynomial
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Citation Context ...ticular zero-dimensional case: Chistov, Grigoriev [19], Canny [17], Gianni, Mora [32], Kobayashi, Fujise, Furukawa [47], Heintz, Roy, Solerno [46], Lakshman, Lazard [54], Renegar [65], Giusti, Heintz =-=[34]-=-, Alonso, Becker, Roy, Wörman [6] and many others. From a practical point of view, the computation of such a representation is always relying on Gröbner basis computations [84], either with a pure lex... |

49 |
Zur Ermittlung eines Objektes aus zwei Perspektiven mit innerer Orientierung,” Sitzungsberichte Österreichische Akademie der Wissenschaften
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(Show Context)
Citation Context ...is. Moreover, in this case our result is deterministic since the number of solutions found is equal to the Bézout number of the system. 8.3 Camera Calibration (Kruppa) The original problem comes from =-=[52]-=- and has been introduced in computer vision in [60]. It is composed of 5 equations in 5 variables. Each equation is a difference of two products of two linear forms. The parameter h is the size of the... |

49 |
Algorithms for solving equations
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(Show Context)
Citation Context ... numerical algorithms is the certification of the result. In [18] a comparison is made between our method and the numerical approach using homotopy and the approximate zero theory introduced by Smale =-=[74, 75]-=-. 2.2 Description of the Computations We present now the actual computations performed by our algorithm in a generic case. Our algorithm is incremental in the number of equations to be solved. Let Si ... |

42 |
Sylvester–habicht sequences and fast cauchy index computation
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- 2001
(Show Context)
Citation Context ...als of degree at most n over a field K can be computed in M(n) arithmetic operations in K [61]. The resultant, the sub-resultants and the interpolation can also be computed within the same complexity =-=[57, 29]-=-. The Schönhage-Strassen algorithm [71] for multiplying two integers of bitsize at most n has a bit-complexity in O(n log(n) log log(n)). The division has the same complexity as the multiplication [73... |

41 |
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(Show Context)
Citation Context ...l to obtain complexity results by many authors, in the particular zero-dimensional case: Chistov, Grigoriev [19], Canny [17], Gianni, Mora [32], Kobayashi, Fujise, Furukawa [47], Heintz, Roy, Solerno =-=[46]-=-, Lakshman, Lazard [54], Renegar [65], Giusti, Heintz [34], Alonso, Becker, Roy, Wörman [6] and many others. From a practical point of view, the computation of such a representation is always relying ... |

35 |
zur Gathen, Parallel arithmetic computations: A survey. Mathematical foundations of computer science
- von
- 1986
(Show Context)
Citation Context ...tegers does not appear in the complexity if the integers are represented by straight-line programs. For exact definitions and elementary properties of the notion of straight-line programs we refer to =-=[77, 82, 76, 44]-=-. A good historical presentation of all these works can be found in [18] and a didactic presentation of the algorithm is in [62]. We recall the main statement of these works: Theorem [37] Let g and f1... |

32 | Converting Bases with the Gröbner Walk
- Collart, Kalkbrener, et al.
- 1997
(Show Context)
Citation Context ...t of view, the computation of such a representation is always relying on Gröbner basis computations [84], either with a pure lexicographical elimination order, or with an algorithm of change of basis =-=[28, 20]-=- or from any basis using a generalization of Newton’s formulæ by Rouillier [66, 67]. In 1995, Giusti, Heintz, Morais and Pardo [36, 64] rediscovered Kronecker’s approach without any prior knowledge of... |

32 |
Die berechnungskomplexitat von elementarsymmetrischen funktionen und von interpolationskoeffizienten
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(Show Context)
Citation Context ...0, 63] of two polynomials of R[T ] of degree at most n can be performed in O(n log(n) log log(n)) arithmetic operations in R. The division of polynomials has the same complexity as the multiplication =-=[11, 78]-=-. The greatest common divisor of two polynomials of degree at most n over a field K can be computed in M(n) arithmetic operations in K [61]. The resultant, the sub-resultants and the interpolation can... |

30 |
Fast computation of Gcds
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- 1973
(Show Context)
Citation Context ...olynomials has the same complexity as the multiplication [11, 78]. The greatest common divisor of two polynomials of degree at most n over a field K can be computed in M(n) arithmetic operations in K =-=[61]-=-. The resultant, the sub-resultants and the interpolation can also be computed within the same complexity [57, 29]. The Schönhage-Strassen algorithm [71] for multiplying two integers of bitsize at mos... |

29 |
How lower and upper complexity bounds meet in elimination theory
- Pardo
- 1995
(Show Context)
Citation Context ...hical elimination order, or with an algorithm of change of basis [28, 20] or from any basis using a generalization of Newton’s formulæ by Rouillier [66, 67]. In 1995, Giusti, Heintz, Morais and Pardo =-=[36, 64]-=- rediscovered Kronecker’s approach without any prior knowledge of it and improved the space complexity but not the running time complexity. A first breakthrough was obtained by Giusti, Hägele, Heintz,... |

26 |
An algorithm for division of power series
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(Show Context)
Citation Context ...29]. The Schönhage-Strassen algorithm [71] for multiplying two integers of bitsize at most n has a bit-complexity in O(n log(n) log log(n)). The division has the same complexity as the multiplication =-=[73]-=-. The greatest common divisor has complexity M(n) [69]. Let R be a unitary ring, the multiplication of two n × n matrices can be done in O(n ω ) arithmetic operations in R. The exponent ω can be taken... |

26 | Probabilistic algorithms for veri of polynomial identities - Schwartz - 1980 |

24 | On the time-space complexity of geometric elimination procedures
- Heintz, Matera, et al.
(Show Context)
Citation Context ... straight-line programs anymore, neither for multivariate polynomials nor for integer numbers. We give a new estimate of the exponents of the complexity of Theorem [37] above improving the results of =-=[45]-=-. One main step of this transformation is obtained by a technique reminiscent of the deforestation [83], that we had already used in [39] to replace straightline programs by an efficient use of specia... |

22 |
On the intrinsic complexity of the arithmetic Nullstellensatz
- Hagele, Morais, et al.
(Show Context)
Citation Context ...utations and Hensel’s lifting (for example see [23, §4.1.1] or [31, §7.4]). The description of the probability of choosing a lucky p is out of the scope of this work but such considerations are as in =-=[40, 41, 42]-=-. The probability of failure of the algorithm given in [45] has been studied using Zippel-Schwartz’s zero test [85, 72] for multivariate polynomials. We could use the same analysis here to quantify th... |

22 | Bounds for traces in complete intersections and degrees in the Nullstellensatz - Sabia, Solernó - 1995 |

21 | On the efficiency of effective Nullstellensätze - Giusti, Heintz, et al. - 1993 |

18 |
Polynomial-time factoring of the multivariable polynomials over a global field, Preprint LOMI E-5-82
- Chistov, Grigoriev
- 1982
(Show Context)
Citation Context ...ound in Macaulay’s book [58]. This representation has been used in computer algebra as a tool to obtain complexity results by many authors, in the particular zero-dimensional case: Chistov, Grigoriev =-=[19]-=-, Canny [17], Gianni, Mora [32], Kobayashi, Fujise, Furukawa [47], Heintz, Roy, Solerno [46], Lakshman, Lazard [54], Renegar [65], Giusti, Heintz [34], Alonso, Becker, Roy, Wörman [6] and many others.... |