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30
A Gröbner free alternative for polynomial system solving
 Journal of Complexity
, 2001
"... 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 ..."
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Cited by 109 (19 self)
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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
On The Complexity Of Computing Determinants
 COMPUTATIONAL COMPLEXITY
, 2001
"... We present new baby steps/giant steps algorithms of asymptotically fast running time for dense matrix problems. Our algorithms compute the determinant, characteristic polynomial, Frobenius normal form and Smith normal form of a dense n n matrix A with integer entries in (n and (n bi ..."
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Cited by 63 (21 self)
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We present new baby steps/giant steps algorithms of asymptotically fast running time for dense matrix problems. Our algorithms compute the determinant, characteristic polynomial, Frobenius normal form and Smith normal form of a dense n n matrix A with integer entries in (n and (n bit operations; here denotes the largest entry in absolute value and the exponent adjustment by "+o(1)" captures additional factors for positive real constants C 1 , C 2 , C 3 . The bit complexity (n results from using the classical cubic matrix multiplication algorithm. Our algorithms are randomized, and we can certify that the output is the determinant of A in a Las Vegas fashion. The second category of problems deals with the setting where the matrix A has elements from an abstract commutative ring, that is, when no divisions in the domain of entries are possible. We present algorithms that deterministically compute the determinant, characteristic polynomial and adjoint of A with n and O(n ) ring additions, subtractions and multiplications.
Greatest Common Divisors of Polynomials Given by StraightLine Programs
 J. ACM
, 1988
"... . F Algorithms on multivariate polynomials represented by straightline programs are developed irst it is shown that most algebraic algorithms can be probabilistically applied to data that is given by y r a straightline computation. Testing such rational numeric data for zero, for instance, is faci ..."
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Cited by 56 (17 self)
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. F Algorithms on multivariate polynomials represented by straightline programs are developed irst it is shown that most algebraic algorithms can be probabilistically applied to data that is given by y r a straightline computation. Testing such rational numeric data for zero, for instance, is facilitated b andom evaluations modulo random prime numbers. Then auxiliary algorithms are constructed that a determine the coefficients of a multivariate polynomial in a single variable. The first main result is an lgorithm that produces the greatest common divisor of the input polynomials, all in straightline r a representation. The second result shows how to find a straightline program for the reduced numerato nd denominator from one for the corresponding rational function. Both the algorithm for that conl c struction and the greatest common divisor algorithm are in random polynomialtime for the usua oefficient fields and output a straightline program, which with controllably high probab...
Fast Polynomial Factorization Over High Algebraic Extensions of Finite Fields
 In Kuchlin [1997
, 1997
"... New algorithms are presented for factoring polynomials of degree n over the finite field of q elements, where q is a power of a fixed prime number. When log q = n 1+a , where a ? 0 is constant, these algorithms are asymptotically faster than previous known algorithms, the fastest of which require ..."
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Cited by 24 (5 self)
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New algorithms are presented for factoring polynomials of degree n over the finite field of q elements, where q is a power of a fixed prime number. When log q = n 1+a , where a ? 0 is constant, these algorithms are asymptotically faster than previous known algorithms, the fastest of which required time \Omega\Gamma n(log q) 2 ), y or \Omega\Gamma n 3+2a ) in this case, which corresponds to the cost of computing x q modulo an n degree polynomial. The new algorithms factor an arbitrary polynomial in time O(n 3+a+o(1) +n 2:69+1:69a ). All measures are in fixed precision operations, that is in bit complexity. Moreover, in the special case where all the irreducible factors have the same degree, the new algorithms run in time O(n 2:69+1:69a ). In particular, one may test a polynomial for irreducibility in O(n 2:69+1:69a ) bit operations. These results generalize to the case where q = p k , where p is a small prime number relative to q. 1 Introduction The expected run...
An Extension of Kedlaya's Algorithm to Hyperelliptic Curves in Characteristic 2
, 2002
"... We present an algorithm for computing the zeta function of an arbitrary hyperelliptic curve over a finite field Fq of characteristic 2, thereby extending the algorithm of Kedlaya for odd characteristic. For a genus g hyperelliptic curve defined over F2 n , the averagecase time complexity is O(g ) a ..."
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Cited by 18 (5 self)
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We present an algorithm for computing the zeta function of an arbitrary hyperelliptic curve over a finite field Fq of characteristic 2, thereby extending the algorithm of Kedlaya for odd characteristic. For a genus g hyperelliptic curve defined over F2 n , the averagecase time complexity is O(g ) and the averagecase space complexity is O(g ), whereas the worstcase time and space complexities are O(g ) and ) respectively.
Acceleration of Euclidean algorithm and rational number reconstruction
 SIAM Journal on Computing
, 2003
"... Abstract. We accelerate the known algorithms for computing a selected entry of the extended Euclidean algorithm for integers and, consequently, for the modular and numerical rational number reconstruction problems. The acceleration is from quadratic to nearly linear time, matching the known complexi ..."
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Cited by 15 (4 self)
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Abstract. We accelerate the known algorithms for computing a selected entry of the extended Euclidean algorithm for integers and, consequently, for the modular and numerical rational number reconstruction problems. The acceleration is from quadratic to nearly linear time, matching the known complexity bound for the integer gcd, which our algorithm computes as a special case.
Variations by complexity theorists on three themes of Euler, . . .
 COMPUTATIONAL COMPLEXITY
, 2005
"... This paper surveys some connections between geometry and complexity. A main role is played by some quantities —degree, Euler characteristic, Betti numbers — associated to algebraic or semialgebraic sets. This role is twofold. On the one hand, lower bounds on the deterministic time (sequential and pa ..."
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Cited by 15 (3 self)
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This paper surveys some connections between geometry and complexity. A main role is played by some quantities —degree, Euler characteristic, Betti numbers — associated to algebraic or semialgebraic sets. This role is twofold. On the one hand, lower bounds on the deterministic time (sequential and parallel) necessary to decide a set S are established as functions of these quantities associated to S. The optimality of some algorithms is obtained as a consequence. On the other hand, the computation of these quantities gives rise to problems which turn out to be hard (or complete) in different complexity classes. These two kind of results thus turn the quantities above into measures of complexity in two quite different ways.
An OutputSensitive Variant of the Baby Steps/Giant Steps Determinant Algorithm
, 2001
"... This paper provides an adaptive version of the unblocked baby steps/giant steps algorithm [20, Section 2]. The result is most easily stated when b # where # is the determinant to be computed and # with 1 is not known. Note that by Hadamard's bound # # n(b + log 2 (n)/2), so # = ..."
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Cited by 14 (2 self)
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This paper provides an adaptive version of the unblocked baby steps/giant steps algorithm [20, Section 2]. The result is most easily stated when b # where # is the determinant to be computed and # with 1 is not known. Note that by Hadamard's bound # # n(b + log 2 (n)/2), so # = 0 covers the worst case. We describe a Monte Carlo algorithm that produces # in (n bit operations, again with standard matrix arithmetic. The corresponding bit complexity of the early termination Gaussian elimination method is 4# , which is always more, and that of the algorithm by [10] is (n 1+1/2
Uniform closure properties of pcomputable functions
 Proc. 18th ACM Symp. Theory Comp
, 1986
"... Valiant [24] introduced the notion of a family of pcomputable polynomials as those multivariate polynomials of polynomiallybounded degree and straightline computation length. He raised the question of whether pcomputable families would be closed under natural mathematical operations and showed t ..."
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Cited by 9 (5 self)
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Valiant [24] introduced the notion of a family of pcomputable polynomials as those multivariate polynomials of polynomiallybounded degree and straightline computation length. He raised the question of whether pcomputable families would be closed under natural mathematical operations and showed that this is true for taking repeated partial derivatives inasingle variable,
Around the numericsymbolic computation of differential Galois groups
, 2006
"... Let L ∈ K(z)[∂] be a linear differential operator, where K is an effective algebraically closed subfield of C. It can be shown that the differential Galois group of L is generated (as a closed algebraic group) by a finite number of monodromy matrices, Stokes matrices and matrices in local exponentia ..."
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Cited by 8 (0 self)
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Let L ∈ K(z)[∂] be a linear differential operator, where K is an effective algebraically closed subfield of C. It can be shown that the differential Galois group of L is generated (as a closed algebraic group) by a finite number of monodromy matrices, Stokes matrices and matrices in local exponential groups. Moreover, there exist fast algorithms for the approximation of the entries of these matrices. In this paper, we present a numericsymbolic algorithm for the computation of the closed algebraic subgroup generated by a finite number of invertible matrices. Using the above results, this yields an algorithm for the computation of differential Galois groups, when computing with a sufficient precision. Even though there is no straightforward way to find a “sufficient precision ” for guaranteeing the correctness of the endresult, it is often possible to check a posteriori whether the endresult is correct. In particular, we present a nonheuristic algorithm for the factorization of linear differential operators. 1.