## Probabilistic Algorithms for Geometric Elimination (1999)

Venue: | in Engineering, Communication and Computing |

Citations: | 12 - 5 self |

### BibTeX

@ARTICLE{Matera99probabilisticalgorithms,

author = {Guillermo Matera},

title = {Probabilistic Algorithms for Geometric Elimination},

journal = {in Engineering, Communication and Computing},

year = {1999},

volume = {6},

pages = {463--520}

}

### OpenURL

### Abstract

We develop probabilistic algorithms that solve problems of geometric elimination theory using small memory resources. These algorithms are obtained by means of the adaptation of a general transformation due to A. Borodin which converts uniform boolean circuit depth into sequential (Turing machine) space. The boolean circuits themselves are developed using techniques based on the computation of a primitive element of a suitable zero-dimensional algebra and diophantine considerations. Our algorithms improve...

### Citations

358 | The Complexity of Boolean Functions
- Wegener
- 1987
(Show Context)
Citation Context ...ination theory, boolean circuits, arithmetic circuits. 1 Introduction We use standard notions and notations for boolean complexity models and boolean complexity classes as can be found in [4], [3] or =-=[63-=-]. We recall that the classes NC i are dened as the set of O(log i n){uniform families of boolean circuits of polynomial size and depth O(log i n) with bounded fan-in. A family of boolean circuit is c... |

285 |
Parallel Algorithms for Shared-Memory Machines
- Karp, Ramachandran
- 1990
(Show Context)
Citation Context ...lized arithmetic circuit Cn computing the inner product P n i=1 x i y i by using two basic operations: the product of two integers and the addition of n integers. As it is well known (see for example =-=[42]-=- or [63]) there exists a family of boolean circuits that computes the product of two integers of logarithmic height h with size O(h 2 ) and depth O(log h) which is uniform in space O(log h). Using the... |

282 |
Parallelism in random access machines
- Fortune, Wyllie
(Show Context)
Citation Context ...the length of the longest path joining an input node with an output node when only nonscalar nodes are considered. In the sequel we are going to use a general argument due to A. Borodin [7] (see also =-=[21]-=-), which allows to turn uniform boolean circuit depth (parallel time) into sequential space. This argument allows us to re{interpret the property of well parallelizability of a boolean function as the... |

239 |
Probabilistic algorithms for sparse polynomials
- Zippel
- 1979
(Show Context)
Citation Context ...n in some points chosen at random in a suitable set. Throughout this work we are going to consider two alternative test methods: the Schwartz{Zippel test (see e.g. [37], Corollary 2.1; see also [60], =-=[64]-=-, [41], [38]) and the Heintz{Schnorr test (see e.g. [36], Theorem 4.4; [45], Corollary 19 or [58]). An important point of our algorithmic method is the representation of algebraic varieties by means o... |

172 |
Structural Complexity I
- Balcázar, Díaz, et al.
- 1995
(Show Context)
Citation Context ...s, elimination theory, boolean circuits, arithmetic circuits. 1 Introduction We use standard notions and notations for boolean complexity models and boolean complexity classes as can be found in [4], =-=[3-=-] or [63]. We recall that the classes NC i are dened as the set of O(log i n){uniform families of boolean circuits of polynomial size and depth O(log i n) with bounded fan-in. A family of boolean circ... |

116 |
On computing the determinant in small parallel time using a small number of processors
- Berkowitz
- 1984
(Show Context)
Citation Context ...elated to the uniformity of the linear algebra circuits we use. Although several linear algebra algorithms are announced to be space{uniform, no proof of such statements is explicitly shown (see e.g. =-=[6]-=- or [8]). This section is devoted to review the linear algebra algorithms we need, giving simple proofs of their space{uniformity based on our Macro expansion lemma. These linear algebra algorithms ar... |

94 | On relating time and space to size and depth
- Borodin
- 1977
(Show Context)
Citation Context ...sis dened as the length of the longest path joining an input node with an output node when only nonscalar nodes are considered. In the sequel we are going to use a general argument due to A. Borodin [=-=7]-=- (see also [21]), which allows to turn uniform boolean circuit depth (parallel time) into sequential space. This argument allows us to re{interpret the property of well parallelizability of a boolean ... |

82 |
On Euclid’s algorithm and the theory of subresultants
- Brown, Traub
- 1971
(Show Context)
Citation Context ... : : ; Xn ]{ multiple of the greatest common divisor of f and g over Q(X 1 ; : : : ; Xn )[T ], which is uniform in space S +O log(dHL) . Proof. We apply techniques based on subresultants (see [15], [=-=10]-=-) in the version of [8]: it turns out that the minimal natural number j such that the j{th principal submatrix P j (f; g) of the Sylvester matrix is nonsingular equals the degree of gcd(f; g) (see The... |

68 |
Vermeidung von divisionen
- Strassen
- 1973
(Show Context)
Citation Context ...nomials F1 F0 ; : : : ; Fm F0 by means of an arithmetic circuit without divisions by nonconstant polynomials of R. For this sake we are going to use the well{known procedure Vermeidung von Divisionen =-=[62]-=- in the version of [45] which allows to control the height of the parameters involved. Lemma 14. Let fC n;m;d;h g n;m;d;h2IN be a family of boolean circuits which evaluates polynomials F 0 ; : : : ; F... |

65 | Grundzüge einer arithmetischen Theorie der algebraischen Grössen - Kronecker - 1882 |

60 | Lower bounds for diophantine approximation
- Giusti, Hägele, et al.
- 1997
(Show Context)
Citation Context ...ation of algebraic varieties by means of primitive element techniques. The idea | which is originally due to Kronecker [46] and has been applied in several papers such as [2], [11], [13], [14], [24], =-=[26], [28-=-], [29], [30], [43], [45], [49], [51], [56] | consists insnding a \pseudo{parameterization" of the variety under consideration which, at the same time, provides valuable algebraic information. Th... |

60 | When polynomial equation systems can be solved fast
- Giusti, Heintz, et al.
- 1995
(Show Context)
Citation Context ... varieties by means of primitive element techniques. The idea | which is originally due to Kronecker [46] and has been applied in several papers such as [2], [11], [13], [14], [24], [26], [28], [29], =-=[30], [43-=-], [45], [49], [51], [56] | consists insnding a \pseudo{parameterization" of the variety under consideration which, at the same time, provides valuable algebraic information. This pseudo{paramete... |

57 |
A computational method for diophantine approximation
- Krick, Pardo
- 1996
(Show Context)
Citation Context ...e are going to consider two alternative test methods: the Schwartz{Zippel test (see e.g. [37], Corollary 2.1; see also [60], [64], [41], [38]) and the Heintz{Schnorr test (see e.g. [36], Theorem 4.4; =-=[45]-=-, Corollary 19 or [58]). An important point of our algorithmic method is the representation of algebraic varieties by means of primitive element techniques. The idea | which is originally due to Krone... |

53 | Zeroes, multiplicities and idempotents for zerodimensional systems
- Alonso, Becker, et al.
- 1996
(Show Context)
Citation Context ...thmic method is the representation of algebraic varieties by means of primitive element techniques. The idea | which is originally due to Kronecker [46] and has been applied in several papers such as =-=[2], [11-=-], [13], [14], [24], [26], [28], [29], [30], [43], [45], [49], [51], [56] | consists insnding a \pseudo{parameterization" of the variety under consideration which, at the same time, provides valu... |

51 | Greatest common divisors of polynomials given by straight-line programs
- Kaltofen
- 1988
(Show Context)
Citation Context ...ome points chosen at random in a suitable set. Throughout this work we are going to consider two alternative test methods: the Schwartz{Zippel test (see e.g. [37], Corollary 2.1; see also [60], [64], =-=[41]-=-, [38]) and the Heintz{Schnorr test (see e.g. [36], Theorem 4.4; [45], Corollary 19 or [58]). An important point of our algorithmic method is the representation of algebraic varieties by means of prim... |

50 | La détermination des points isolés et de la dimension d'une variété algebrique peut se faire en temps polynomial
- Giusti, Heintz
- 1991
(Show Context)
Citation Context ...of algebraic varieties by means of primitive element techniques. The idea | which is originally due to Kronecker [46] and has been applied in several papers such as [2], [11], [13], [14], [24], [26], =-=[28], [29-=-], [30], [43], [45], [49], [51], [56] | consists insnding a \pseudo{parameterization" of the variety under consideration which, at the same time, provides valuable algebraic information. This pse... |

49 | Algebraic solution of systems of polynomial equations using Gröbner bases. In Applied algebra, algebraic algorithms and error-correcting codes (Menorca
- Gianni, Mora
- 1987
(Show Context)
Citation Context ...resentation of algebraic varieties by means of primitive element techniques. The idea | which is originally due to Kronecker [46] and has been applied in several papers such as [2], [11], [13], [14], =-=[24], [26-=-], [28], [29], [30], [43], [45], [49], [51], [56] | consists insnding a \pseudo{parameterization" of the variety under consideration which, at the same time, provides valuable algebraic informati... |

49 |
Théorèmes de Bertini et Applications
- Jouanolou
- 1983
(Show Context)
Citation Context ...om the set f1; : : : ; d 2n g n with probability of success greater than 1 1 d n 2 . This is achieved by means of a combination of Schwartz{Zippel test and an eective version of Bertini Theorem (cf. [=-=40]-=-) in the version of [45]. Furthermore, there exists a linear change of coordinates (X 1 ; : : : ; Xn ) ! (Y 1 ; : : : ; Yn ) such that the variables Y 1 ; : : : ; Yn are in Noether position with respe... |

48 |
Résolution des systèmes d'équations algébriques
- Lazard
- 1981
(Show Context)
Citation Context ...rimitive element techniques. The idea | which is originally due to Kronecker [46] and has been applied in several papers such as [2], [11], [13], [14], [24], [26], [28], [29], [30], [43], [45], [49], =-=[51], [56-=-] | consists insnding a \pseudo{parameterization" of the variety under consideration which, at the same time, provides valuable algebraic information. This pseudo{parameterization is given by a c... |

41 | Fast parallel matrix and gcd computations
- Borodin, Gathen, et al.
- 1982
(Show Context)
Citation Context ...to the uniformity of the linear algebra circuits we use. Although several linear algebra algorithms are announced to be space{uniform, no proof of such statements is explicitly shown (see e.g. [6] or =-=[8]-=-). This section is devoted to review the linear algebra algorithms we need, giving simple proofs of their space{uniformity based on our Macro expansion lemma. These linear algebra algorithms are gener... |

39 |
zur Gathen. Parallel linear algebra. This volume
- von
(Show Context)
Citation Context ...y of boolean circuit is called S(n){space uniform if its standard encoding can be built using deterministic space S(n). We will also use the model of division{free arithmetic circuits (see e.g. [22], =-=[23-=-] or [1]). Let R be a ring. An arithmetic circuitsover R is a directed acyclic graph (dag for short) (), where all nodes have bounded indegree of either 0 or 2. The nodes of indegree 0 representing in... |

33 |
zur Gathen. Parallel arithmetic computations: a survey
- von
- 1986
(Show Context)
Citation Context ... family of boolean circuit is called S(n){space uniform if its standard encoding can be built using deterministic space S(n). We will also use the model of division{free arithmetic circuits (see e.g. =-=[22-=-], [23] or [1]). Let R be a ring. An arithmetic circuitsover R is a directed acyclic graph (dag for short) (), where all nodes have bounded indegree of either 0 or 2. The nodes of indegree 0 represent... |

29 |
How lower and upper complexity bounds meet in elimination theory
- Pardo
- 1995
(Show Context)
Citation Context ...tral idea of the method of [53] consists in the reduction of elimination problems to linear algebra computations. These reductions rely on suitable versions of eective Nullstellensatze (cf. [35] or [5=-=8]-=-). Unfortunately, the matrices occurring in these reductions have exponential size, mainly due to the syntactical aspect of the kind of problems under consideration. By this syntactical aspect we refe... |

28 |
Testing polynomials which are easy to compute
- Heintz, Schnorr
- 1982
(Show Context)
Citation Context ...oughout this work we are going to consider two alternative test methods: the Schwartz{Zippel test (see e.g. [37], Corollary 2.1; see also [60], [64], [41], [38]) and the Heintz{Schnorr test (see e.g. =-=[36]-=-, Theorem 4.4; [45], Corollary 19 or [58]). An important point of our algorithmic method is the representation of algebraic varieties by means of primitive element techniques. The idea | which is orig... |

26 |
Subexponential time solving systems of algebraic equations
- Chistov, Grigoriev
- 1983
(Show Context)
Citation Context ...he representation of algebraic varieties by means of primitive element techniques. The idea | which is originally due to Kronecker [46] and has been applied in several papers such as [2], [11], [13], =-=[14], [24-=-], [26], [28], [29], [30], [43], [45], [49], [51], [56] | consists insnding a \pseudo{parameterization" of the variety under consideration which, at the same time, provides valuable algebraic inf... |

26 |
Probabilistic algorithms for veri of polynomial identities
- Schwartz
- 1980
(Show Context)
Citation Context ...eration in some points chosen at random in a suitable set. Throughout this work we are going to consider two alternative test methods: the Schwartz{Zippel test (see e.g. [37], Corollary 2.1; see also =-=[60]-=-, [64], [41], [38]) and the Heintz{Schnorr test (see e.g. [36], Theorem 4.4; [45], Corollary 19 or [58]). An important point of our algorithmic method is the representation of algebraic varieties by m... |

24 |
Multivariate Bezoutians, Kronecker Symbol and Eisenbud-Levine formula
- Becker, Cardinal, et al.
(Show Context)
Citation Context ...Hilbert function of a suitable graduate ring as in [51], [9] or [28] (see also [61]) or the division modulo a complete intersection ideal by means of a trace formula as in [20] or [45] (see also [2], =-=[5]-=-, [12] and [59]). Applying all these tools we build arithmetic circuits which have certain nodes that are generated randomly. Then we translate these arithmetic circuits into boolean circuits followin... |

23 |
The structure of polynomial ideals and Gröbner bases
- Dubé
- 1990
(Show Context)
Citation Context ...4 log 2 (shd) space and (shd) O n 4 log(shd) time bounds of [53], and the (shd) O(n 2 ) space and (shd) O(n 2 ) time bounds of the algorithms based on Grobner basis computations (see [25], [44], [17] and [55]). Let us mention in this context that the probabilistic (hsd n ) O(1) space and (hsd n ) O(1) time bounds of the algorithms of [45] and the probabilistic (nh) O(log n) space and (nh) O(log... |

23 |
On the intrinsic complexity of elimination theory
- Heintz, Morgenstern
- 1993
(Show Context)
Citation Context ... The central idea of the method of [53] consists in the reduction of elimination problems to linear algebra computations. These reductions rely on suitable versions of eective Nullstellensatze (cf. [3=-=5]-=- or [58]). Unfortunately, the matrices occurring in these reductions have exponential size, mainly due to the syntactical aspect of the kind of problems under consideration. By this syntactical aspect... |

20 |
Algorithmes disons rapides pour la décomposition d'une variété algébrique en composantes irréductibles et équidimensionnelles
- Giusti, Heintz
- 1990
(Show Context)
Citation Context ...alar depth O(n 2 log d), and which has the following property: any vectors2 ZZ n(i+1) with P i ( ) 6= 0 yields the coecients of i hyperplanes satisfying conditions i), ii) and iii) above. Proof. From =-=[27]-=- we deduce that each component V i can be described as the set of common zeros of certain polynomials G (i) 1 ; : : : ; G (i) t i in ZZ[X 1 ; : : : ; Xn ] of degree bounded by d n . We introduce new v... |

18 |
The membership problem for unmixed polynomial ideals is solvable in single exponential time
- Dickenstein, Fitchas, et al.
- 1991
(Show Context)
Citation Context ... ) ! (Y 1 ; : : : ; Yn ) such that the variables Y 1 ; : : : ; Yn are in Noether position with respect to the variety V i := V (F 0 1 ; : : : ; F 0 i ) for i = 1; : : : ; t. Applying the arguments of =-=[16]-=- or [45], we deduce that the entries of the matrix that performs this linear change of variables can be generated by means of the Schwartz{Zippel test in such a way that their logarithmic height is of... |

18 |
On the complexity of zero-dimensional algebraic systems
- Lakshman, Lazard
- 1990
(Show Context)
Citation Context ...s of primitive element techniques. The idea | which is originally due to Kronecker [46] and has been applied in several papers such as [2], [11], [13], [14], [24], [26], [28], [29], [30], [43], [45], =-=[49], [51-=-], [56] | consists insnding a \pseudo{parameterization" of the variety under consideration which, at the same time, provides valuable algebraic information. This pseudo{parameterization is given ... |

16 |
Sur la complexité du théorème des zéros
- Fitchas, Giusti, et al.
- 1995
(Show Context)
Citation Context ...ts on the regularity of the Hilbert function of a suitable graduate ring as in [51], [9] or [28] (see also [61]) or the division modulo a complete intersection ideal by means of a trace formula as in =-=[20]-=- or [45] (see also [2], [5], [12] and [59]). Applying all these tools we build arithmetic circuits which have certain nodes that are generated randomly. Then we translate these arithmetic circuits int... |

16 |
Le rôle des structures de données dans les problèmes d’élimination
- Giusti, Heintz, et al.
- 1223
(Show Context)
Citation Context ...ilistic (hsd n ) O(1) space and (hsd n ) O(1) time bounds of the algorithms of [45] and the probabilistic (nh) O(log n) space and (nh) O(log n) time bounds of the algorithms of [29], [26], [57] and [3=-=1-=-] (heresand are geometric parameters that can be estimated, in worst case, by d n and hd n respectively) have a better time performance than our procedures but require much more space than ours. Let ... |

16 | Some complexity results for polynomial ideals
- Mayr
- 1997
(Show Context)
Citation Context ...shd) space and (shd) O n 4 log(shd) time bounds of [53], and the (shd) O(n 2 ) space and (shd) O(n 2 ) time bounds of the algorithms based on Grobner basis computations (see [25], [44], [17] and [55]). Let us mention in this context that the probabilistic (hsd n ) O(1) space and (hsd n ) O(1) time bounds of the algorithms of [45] and the probabilistic (nh) O(log n) space and (nh) O(log n) time ... |

12 | Bounds for the Hilbert function of polynomial ideals and for the degrees in the Nullstellensatz
- Sombra
- 1997
(Show Context)
Citation Context ...n to the matrices occurring during our procedures. This method of reduction relies on arguments on the regularity of the Hilbert function of a suitable graduate ring as in [51], [9] or [28] (see also =-=[61]-=-) or the division modulo a complete intersection ideal by means of a trace formula as in [20] or [45] (see also [2], [5], [12] and [59]). Applying all these tools we build arithmetic circuits which ha... |

11 |
De and fast quanti elimination in algebraically closed
- Heintz
- 1983
(Show Context)
Citation Context ...he decomposition of V in equidimensional components, where V i is empty or an equidimensional variety of dimension i for i = 0; : : : ; r. Let 0 i r and suppose that V i is nonempty. As proved in [34], if we choose i generic hyperplanes H (i) 1 ; : : : ; H (i) i , the following conditions are satised: i) V i \ H (i) 1 \ \ H (i) i is a zero{dimensional variety of cardinality deg(V i ). ii) V... |

8 |
Some algebraic and geometric problems in PSPACE
- Canny
- 1988
(Show Context)
Citation Context ... method is the representation of algebraic varieties by means of primitive element techniques. The idea | which is originally due to Kronecker [46] and has been applied in several papers such as [2], =-=[11], [13-=-], [14], [24], [26], [28], [29], [30], [43], [45], [49], [51], [56] | consists insnding a \pseudo{parameterization" of the variety under consideration which, at the same time, provides valuable a... |

8 |
Generic local structure of the morphisms in commutative algebra
- Iversen
- 1973
(Show Context)
Citation Context ... unique trace 2 B such that the following identity holds in B: 1 = X m (am ) c m The main property of the trace is known as the \trace formula" (\Tate's trace formula" of [48, Appendix F=-=] or -=-[39] being special case of it). The trace formula is the following statement: for any G 2 R[X 1 ; : : : ; Xn ] the identity G = X m (G am ) c m (5) holds true in B. Notice that the polynomial P m (G... |

8 | Membership in polynomial ideals over Q is exponential space complete - MAYR - 1989 |

7 |
Solving systems of algebraic equations by a general elimination method
- Kobayashi, Fujise, et al.
- 1988
(Show Context)
Citation Context ...ties by means of primitive element techniques. The idea | which is originally due to Kronecker [46] and has been applied in several papers such as [2], [11], [13], [14], [24], [26], [28], [29], [30], =-=[43], [45-=-], [49], [51], [56] | consists insnding a \pseudo{parameterization" of the variety under consideration which, at the same time, provides valuable algebraic information. This pseudo{parameterizati... |

7 | Exponential space computation of Gröbner bases
- Kühnle, Mayr
- 1996
(Show Context)
Citation Context ...ks in space O(log i n). Our intention is to adapt this idea to the context of geometric elimination theory. Thesrst systematic attempts in this direction were made in [53] and [52] (see also [54] and =-=[47]-=- in the context of algebraic elimination theory), where deterministic algorithms were designed for the resolution of selected problems of geometric elimination. These algorithms, although very ecient ... |

6 |
Sobre la complejidad en espacio y tiempo de la eliminacion geometrica
- Matera
- 1997
(Show Context)
Citation Context ...tial algorithm which works in space O(log i n). Our intention is to adapt this idea to the context of geometric elimination theory. Thesrst systematic attempts in this direction were made in [53] and =-=[52]-=- (see also [54] and [47] in the context of algebraic elimination theory), where deterministic algorithms were designed for the resolution of selected problems of geometric elimination. These algorithm... |

6 |
Resolucion e de sistemas de ecuaciones polinomiales
- Morais
- 1997
(Show Context)
Citation Context ...he probabilistic (hsd n ) O(1) space and (hsd n ) O(1) time bounds of the algorithms of [45] and the probabilistic (nh) O(log n) space and (nh) O(log n) time bounds of the algorithms of [29], [26], [5=-=7-=-] and [31] (heresand are geometric parameters that can be estimated, in worst case, by d n and hd n respectively) have a better time performance than our procedures but require much more space than o... |

5 |
Equivalence of straight-line programs
- Ibarra, Moran
- 1983
(Show Context)
Citation Context ...e two polynomials under consideration in some points chosen at random in a suitable set. Throughout this work we are going to consider two alternative test methods: the Schwartz{Zippel test (see e.g. =-=[37]-=-, Corollary 2.1; see also [60], [64], [41], [38]) and the Heintz{Schnorr test (see e.g. [36], Theorem 4.4; [45], Corollary 19 or [58]). An important point of our algorithmic method is the representati... |

5 |
Membership problem, representation problem and the computation of the radical for one-dimensional ideals
- Krick, Logar
- 1990
(Show Context)
Citation Context ...) O n 4 log 2 (shd) space and (shd) O n 4 log(shd) time bounds of [53], and the (shd) O(n 2 ) space and (shd) O(n 2 ) time bounds of the algorithms based on Grobner basis computations (see [25], [44], [17] and [55]). Let us mention in this context that the probabilistic (hsd n ) O(1) space and (hsd n ) O(1) time bounds of the algorithms of [45] and the probabilistic (nh) O(log n) space and (nh)... |

4 |
Sur le degré des relations entre polynômes
- Briançon
- 1982
(Show Context)
Citation Context ...ethod of size reduction to the matrices occurring during our procedures. This method of reduction relies on arguments on the regularity of the Hilbert function of a suitable graduate ring as in [51], =-=[9]-=- or [28] (see also [61]) or the division modulo a complete intersection ideal by means of a trace formula as in [20] or [45] (see also [2], [5], [12] and [59]). Applying all these tools we build arith... |

4 |
Some eectivity problems in polynomial ideal theory
- Giusti
- 1984
(Show Context)
Citation Context ...aneous) O n 4 log 2 (shd) space and (shd) O n 4 log(shd) time bounds of [53], and the (shd) O(n 2 ) space and (shd) O(n 2 ) time bounds of the algorithms based on Grobner basis computations (see [25=-=-=-], [44], [17] and [55]). Let us mention in this context that the probabilistic (hsd n ) O(1) space and (hsd n ) O(1) time bounds of the algorithms of [45] and the probabilistic (nh) O(log n) space and... |

4 |
Probabilistic Algorithms and Straightline Programs for some rank decision problems
- Ibarra, Moran, et al.
- 1981
(Show Context)
Citation Context ...ints chosen at random in a suitable set. Throughout this work we are going to consider two alternative test methods: the Schwartz{Zippel test (see e.g. [37], Corollary 2.1; see also [60], [64], [41], =-=[38]-=-) and the Heintz{Schnorr test (see e.g. [36], Theorem 4.4; [45], Corollary 19 or [58]). An important point of our algorithmic method is the representation of algebraic varieties by means of primitive ... |

3 |
Dualité et algorithmes itératives pour la solution des systèmes polynomiaux, Thèse, U. de Rennes I
- Cardinal
- 1993
(Show Context)
Citation Context ...rt function of a suitable graduate ring as in [51], [9] or [28] (see also [61]) or the division modulo a complete intersection ideal by means of a trace formula as in [20] or [45] (see also [2], [5], =-=[12]-=- and [59]). Applying all these tools we build arithmetic circuits which have certain nodes that are generated randomly. Then we translate these arithmetic circuits into boolean circuits following a ge... |