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61
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 82 (16 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
Robust PCPs of Proximity, Shorter PCPs and Applications to Coding
 in Proc. 36th ACM Symp. on Theory of Computing
, 2004
"... We continue the study of the tradeo between the length of PCPs and their query complexity, establishing the following main results (which refer to proofs of satis ability of circuits of size n): 1. We present PCPs of length exp( ~ O(log log n) ) n that can be veri ed by making o(log log n) ..."
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Cited by 80 (25 self)
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We continue the study of the tradeo between the length of PCPs and their query complexity, establishing the following main results (which refer to proofs of satis ability of circuits of size n): 1. We present PCPs of length exp( ~ O(log log n) ) n that can be veri ed by making o(log log n) Boolean queries.
Fast parallel circuits for the quantum Fourier transform
 PROCEEDINGS 41ST ANNUAL SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE (FOCS’00)
, 2000
"... We give new bounds on the circuit complexity of the quantum Fourier transform (QFT). We give an upper bound of O(log n + log log(1/ε)) on the circuit depth for computing an approximation of the QFT with respect to the modulus 2 n with error bounded by ε. Thus, even for exponentially small error, our ..."
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Cited by 55 (3 self)
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We give new bounds on the circuit complexity of the quantum Fourier transform (QFT). We give an upper bound of O(log n + log log(1/ε)) on the circuit depth for computing an approximation of the QFT with respect to the modulus 2 n with error bounded by ε. Thus, even for exponentially small error, our circuits have depth O(log n). The best previous depth bound was O(n), even for approximations with constant error. Moreover, our circuits have size O(n log(n/ε)). We also give an upper bound of O(n(log n) 2 log log n) on the circuit size of the exact QFT modulo 2 n, for which the best previous bound was O(n 2). As an application of the above depth bound, we show that Shor’s factoring algorithm may be based on quantum circuits with depth only O(log n) and polynomialsize, in combination with classical polynomialtime pre and postprocessing. In the language of computational complexity, this implies that factoring is in the complexity class ZPP BQNC, where BQNC is the class of problems computable with boundederror probability by quantum circuits with polylogarithmic depth and polynomial size. Finally, we prove an Ω(log n) lower bound on the depth complexity of approximations of the
Faster Integer Multiplication
 STOC'07
, 2007
"... For more than 35 years, the fastest known method for integer multiplication has been the SchönhageStrassen algorithm running in time O(n log n log log n). Under certain restrictive conditions there is a corresponding Ω(n log n) lower bound. The prevailing conjecture has always been that the complex ..."
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Cited by 41 (0 self)
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For more than 35 years, the fastest known method for integer multiplication has been the SchönhageStrassen algorithm running in time O(n log n log log n). Under certain restrictive conditions there is a corresponding Ω(n log n) lower bound. The prevailing conjecture has always been that the complexity of an optimal algorithm is Θ(n log n). We present a major step towards closing the gap from above by presenting an algorithm running in time n log n 2 O(log ∗ n). The main result is for boolean circuits as well as for multitape Turing machines, but it has consequences to other models of computation as well.
Fast arithmetic for triangular sets: from theory to practice
 ISSAC'07
, 2007
"... We study arithmetic operations for triangular families of polynomials, concentrating on multiplication in dimension zero. By a suitable extension of fast univariate Euclidean division, we obtain theoretical and practical improvements over a direct recursive approach; for a family of special cases, ..."
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Cited by 30 (25 self)
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We study arithmetic operations for triangular families of polynomials, concentrating on multiplication in dimension zero. By a suitable extension of fast univariate Euclidean division, we obtain theoretical and practical improvements over a direct recursive approach; for a family of special cases, we reach quasilinear complexity. The main outcome we have in mind is the acceleration of higherlevel algorithms, by interfacing our lowlevel implementation with languages such as AXIOM or Maple. We show the potential for huge speedups, by comparing two AXIOM implementations of van Hoeij and Monagan's modular GCD algorithm.
Fast Computation of Special Resultants
, 2006
"... We propose fast algorithms for computing composed products and composed sums, as well as diamond products of univariate polynomials. These operations correspond to special multivariate resultants, that we compute using power sums of roots of polynomials, by means of their generating series. ..."
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Cited by 20 (8 self)
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We propose fast algorithms for computing composed products and composed sums, as well as diamond products of univariate polynomials. These operations correspond to special multivariate resultants, that we compute using power sums of roots of polynomials, by means of their generating series.
An O(2 n ) algorithm for graph coloring and other partitioning problems via inclusionexclusion
 in Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2006), IEEE
, 2006
"... We use the principle of inclusion and exclusion, combined with polynomial time segmentation and fast Möbius transform, to solve the generic problem of summing or optimizing over the partitions of n elements into a given number of weighted subsets. This problem subsumes various classical graph partit ..."
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Cited by 18 (1 self)
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We use the principle of inclusion and exclusion, combined with polynomial time segmentation and fast Möbius transform, to solve the generic problem of summing or optimizing over the partitions of n elements into a given number of weighted subsets. This problem subsumes various classical graph partitioning problems, such as graph coloring, domatic partitioning, and MAX kCUT, aswell as machine learning problems like decision graph learning and modelbased data clustering. Our algorithm runs in O ∗ (2 n) time, thus substantially improving on the usual O ∗ (3 n)time dynamic programming algorithm; the notation O ∗ suppresses factors polynomial in n. This result improves, e.g., Byskov’s recent record for graph coloring from O ∗ (2.4023 n) to O ∗ (2 n). We note that twenty five years ago, R. M. Karp used inclusion–exclusion in a similar fashion to reduce the space requirement of the usual dynamic programming algorithms from exponential to polynomial. 1.
Change of ordering for regular chains in positive dimension
 IN ILIAS S. KOTSIREAS, EDITOR, MAPLE CONFERENCE 2006
, 2006
"... We discuss changing the variable ordering for a regular chain in positive dimension. This quite general question has applications going from implicitization problems to the symbolic resolution of some systems of differential algebraic equations. We propose a modular method, reducing the problem to d ..."
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Cited by 16 (7 self)
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We discuss changing the variable ordering for a regular chain in positive dimension. This quite general question has applications going from implicitization problems to the symbolic resolution of some systems of differential algebraic equations. We propose a modular method, reducing the problem to dimension zero and using NewtonHensel lifting techniques. The problems raised by the choice of the specialization points, the lack of the (crucial) information of what are the free and algebraic variables for the new ordering, and the efficiency regarding the other methods are discussed. Strong hypotheses (but not unusual) for the initial regular chain are required. Change of ordering in dimension zero is taken as a subroutine.
FAST COMPUTATION OF POWER SERIES SOLUTIONS OF SYSTEMS OF DIFFERENTIAL EQUATIONS
, 2006
"... We propose new algorithms for the computation of the first N terms of a vector (resp. a basis) of power series solutions of a linear system of differential equations at an ordinary point, using a number of arithmetic operations which is quasilinear with respect to N. Similar results are also given ..."
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Cited by 13 (3 self)
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We propose new algorithms for the computation of the first N terms of a vector (resp. a basis) of power series solutions of a linear system of differential equations at an ordinary point, using a number of arithmetic operations which is quasilinear with respect to N. Similar results are also given in the nonlinear case. This extends previous results obtained by Brent and Kung for scalar differential equations of order one and two.