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57
On Fast Multiplication of Polynomials Over Arbitrary Algebras
 Acta Informatica
, 1991
"... this paper we generalize the wellknown SchonhageStrassen algorithm for multiplying large integers to an algorithm for multiplying polynomials with coefficients from an arbitrary, not necessarily commutative, not necessarily associative, algebra A. Our main result is an algorithm to multiply polyno ..."
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Cited by 149 (6 self)
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this paper we generalize the wellknown SchonhageStrassen algorithm for multiplying large integers to an algorithm for multiplying polynomials with coefficients from an arbitrary, not necessarily commutative, not necessarily associative, algebra A. Our main result is an algorithm to multiply polynomials of degree ! n in
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 84 (28 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.
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 (17 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
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 28 (23 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.
Computing Frobenius Maps And Factoring Polynomials
 Comput. Complexity
, 1992
"... . A new probabilistic algorithm for factoring univariate polynomials over finite fields is presented. To factor a polynomial of degree n over F q , the number of arithmetic operations in F q is O((n 2 +n log q) \Delta (log n) 2 loglog n). The main technical innovation is a new way to compute F ..."
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Cited by 28 (2 self)
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. A new probabilistic algorithm for factoring univariate polynomials over finite fields is presented. To factor a polynomial of degree n over F q , the number of arithmetic operations in F q is O((n 2 +n log q) \Delta (log n) 2 loglog n). The main technical innovation is a new way to compute Frobenius and trace maps in the ring of polynomials modulo the polynomial to be factored. Subject classifications. 68Q40; 11Y16, 12Y05. 1. Introduction We consider the problem of factoring a univariate polynomial over a finite field. This problem plays a central role in computational algebra. Indeed, many of the efficient algorithms for factoring univariate and multivariate polynomials over finite fields, the field of rational numbers, and finite extensions of the rationals solve as a subproblem the problem of factoring univariate polynomials over finite fields (Kaltofen 1990). This problem also has important applications in number theory (Buchmann 1990), coding theory (Berlekamp 1968), and ...
Multidigit Multiplication For Mathematicians
, 2001
"... This paper surveys techniques for multiplying elements of various commutative rings. It covers Karatsuba multiplication, dual Karatsuba multiplication, Toom multiplication, dual Toom multiplication, the FFT trick, the twisted FFT trick, the splitradix FFT trick, Good's trick, the SchönhageStr ..."
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Cited by 27 (9 self)
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This paper surveys techniques for multiplying elements of various commutative rings. It covers Karatsuba multiplication, dual Karatsuba multiplication, Toom multiplication, dual Toom multiplication, the FFT trick, the twisted FFT trick, the splitradix FFT trick, Good's trick, the SchönhageStrassen trick, Schönhage's trick, Nussbaumer's trick, the cyclic SchönhageStrassen trick, and the CantorKaltofen theorem. It emphasizes the underlying ring homomorphisms.
Linear recurrences with polynomial coefficients and computation of the CartierManin operator on hyperelliptic curves
 In International Conference on Finite Fields and Applications (Toulouse
, 2004
"... Abstract. We study the complexity of computing one or several terms (not necessarily consecutive) in a recurrence with polynomial coefficients. As applications, we improve the best currently known upper bounds for factoring integers deterministically and for computing the Cartier–Manin operator of h ..."
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Cited by 22 (8 self)
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Abstract. We study the complexity of computing one or several terms (not necessarily consecutive) in a recurrence with polynomial coefficients. As applications, we improve the best currently known upper bounds for factoring integers deterministically and for computing the Cartier–Manin operator of hyperelliptic curves.
On the Deterministic Complexity of Factoring Polynomials over Finite Fields
 Inform. Process. Lett
, 1990
"... . We present a new deterministic algorithm for factoring polynomials over Z p of degree n. We show that the worstcase running time of our algorithm is O(p 1=2 (log p) 2 n 2+ffl ), which is faster than the running times of previous deterministic algorithms with respect to both n and p. We also ..."
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Cited by 21 (4 self)
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. We present a new deterministic algorithm for factoring polynomials over Z p of degree n. We show that the worstcase running time of our algorithm is O(p 1=2 (log p) 2 n 2+ffl ), which is faster than the running times of previous deterministic algorithms with respect to both n and p. We also show that our algorithm runs in polynomial time for all but at most an exponentially small fraction of the polynomials of degree n over Z p . Specifically, we prove that the fraction of polynomials of degree n over Z p for which our algorithm fails to halt in time O((log p) 2 n 2+ffl ) is O((n log p) 2 =p). Consequently, the averagecase running time of our algorithm is polynomial in n and log p. Keywords: factorization, finite fields, irreducible polynomials. This research was supported by NSF grants DCR8504485 and DCR8552596. Appeared in Information Processing Letters 33, pp. 261267, 1990. An preliminary version of this paper appeared as University of WisconsinMadison, Comput...
Computing Parametric Geometric Resolutions
, 2001
"... Given a polynomial system of n equations in n unknowns that depends on some parameters, we de ne the notion of parametric geometric resolution as a means to represent some generic solutions in terms of the parameters. The coefficients of this resolution are rational functions of the parameters; we f ..."
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Cited by 20 (7 self)
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Given a polynomial system of n equations in n unknowns that depends on some parameters, we de ne the notion of parametric geometric resolution as a means to represent some generic solutions in terms of the parameters. The coefficients of this resolution are rational functions of the parameters; we first show that their degree is bounded by the Bézout number d n , where d is a bound on the degrees of the input system. We then present a probabilistic algorithm to compute such a resolution; in short, its complexity is polynomial in the size of the output and the probability of success is controlled by a quantity polynomial in the Bézout number. We present several applications of this process, to computations in the Jacobian of hyperelliptic curves and to questions of real geometry.
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 18 (7 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.