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THE COMPLETE GENERATING FUNCTION FOR GESSEL WALKS IS ALGEBRAIC
"... Gessel walks are lattice walks in the quarter plane N2 which start at the origin (0, 0) ∈ N2 and consist only of steps chosen from the set {←, ↙, ↗, →}. We prove that if g(n; i, j) denotes the number of Gessel walks of length n which end at the point (i, j) ∈ N2, then the trivariate generating ser ..."
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Cited by 27 (7 self)
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Gessel walks are lattice walks in the quarter plane N2 which start at the origin (0, 0) ∈ N2 and consist only of steps chosen from the set {←, ↙, ↗, →}. We prove that if g(n; i, j) denotes the number of Gessel walks of length n which end at the point (i, j) ∈ N2, then the trivariate generating series G(t; x, y) = X g(n; i, j)x i y j t n is an algebraic function. n,i,j≥0 1.
Lifting and recombination techniques for absolute factorization
 J. Complexity
, 2007
"... Abstract. In the vein of recent algorithmic advances in polynomial factorization based on lifting and recombination techniques, we present new faster algorithms for computing the absolute factorization of a bivariate polynomial. The running time of our probabilistic algorithm is less than quadratic ..."
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Cited by 14 (7 self)
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Abstract. In the vein of recent algorithmic advances in polynomial factorization based on lifting and recombination techniques, we present new faster algorithms for computing the absolute factorization of a bivariate polynomial. The running time of our probabilistic algorithm is less than quadratic in the dense size of the polynomial to be factored.
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 12 (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.
Fast construction of irreducible polynomials over finite fields (version of 22 Apr
, 2009
"... We present a randomized algorithm that on input a finite field K with q elements and a positive integer d outputs a degree d irreducible polynomial in K[x]. The running time is d 1+o(1) × (log q) 5+o(1) elementary operations. The o(1) in d 1+o(1) is a function of d that tends to zero when d tends to ..."
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Cited by 3 (0 self)
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We present a randomized algorithm that on input a finite field K with q elements and a positive integer d outputs a degree d irreducible polynomial in K[x]. The running time is d 1+o(1) × (log q) 5+o(1) elementary operations. The o(1) in d 1+o(1) is a function of d that tends to zero when d tends to infinity. And the o(1) in (log q) 5+o(1) is a function of q that tends to zero when q tends to infinity. In particular, the complexity is quasilinear in the degree d. 1
Genus 2 point counting over prime fields
"... For counting points of jacobians of genus 2 curves over a large prime field, the best known approach is essentially an extension of Schoof’s genus 1 algorithm. We propose various practical improvements to this method and illustrate them with a large scale computation: we counted hundreds of curves, ..."
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Cited by 1 (1 self)
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For counting points of jacobians of genus 2 curves over a large prime field, the best known approach is essentially an extension of Schoof’s genus 1 algorithm. We propose various practical improvements to this method and illustrate them with a large scale computation: we counted hundreds of curves, until one was found that is suitable for cryptographic use, with a stateoftheart security level of approximately 2 128 and desirable speed properties. This curve and its quadratic twist have a Jacobian group whose order is 16 times a prime. Key words: Point couting, hyperelliptic curves, SchoofPila algorithm.
Homotopy techniques for multiplication modulo triangular sets
"... We study the cost of multiplication modulo triangular families of polynomials. Following previous work by Li, Moreno Maza and Schost, we propose an algorithm that relies on homotopy and fast evaluationinterpolation techniques. We obtain a quasilinear time complexity for substantial families of exa ..."
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We study the cost of multiplication modulo triangular families of polynomials. Following previous work by Li, Moreno Maza and Schost, we propose an algorithm that relies on homotopy and fast evaluationinterpolation techniques. We obtain a quasilinear time complexity for substantial families of examples, for which no such result was known before. Applications are given to notably addition of algebraic numbers in small characteristic.
THE COMPLETE GENERATING FUNCTION FOR GESSEL WALKS IS ALGEBRAIC ALIN BOSTAN AND MANUEL KAUERS, WITH AN APPENDIX BY
"... Abstract. Gessel walks are lattice walks in the quarterplane N2 which start at the origin (0, 0) ∈ N2 and consist only of steps chosen from the set {←, ↙, ↗, →}. Weprovethatifg(n; i, j) denotes the number of Gessel walks of length n which end at the point (i, j) ∈ N2, then the trivariate generati ..."
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Abstract. Gessel walks are lattice walks in the quarterplane N2 which start at the origin (0, 0) ∈ N2 and consist only of steps chosen from the set {←, ↙, ↗, →}. Weprovethatifg(n; i, j) denotes the number of Gessel walks of length n which end at the point (i, j) ∈ N2, then the trivariate generating series G(t; x, y) = g(n; i, j)x n,i,j≥0 i y j t n is an algebraic function. 1.