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16
Counting Points on Hyperelliptic Curves over Finite Fields
"... . We describe some algorithms for computing the cardinality of hyperelliptic curves and their Jacobians over finite fields. They include several methods for obtaining the result modulo small primes and prime powers, in particular an algorithm `a la Schoof for genus 2 using Cantor 's division pol ..."
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Cited by 59 (7 self)
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. We describe some algorithms for computing the cardinality of hyperelliptic curves and their Jacobians over finite fields. They include several methods for obtaining the result modulo small primes and prime powers, in particular an algorithm `a la Schoof for genus 2 using Cantor 's division polynomials. These are combined with a birthday paradox algorithm to calculate the cardinality. Our methods are practical and we give actual results computed using our current implementation. The Jacobian groups we handle are larger than those previously reported in the literature. Introduction In recent years there has been a surge of interest in algorithmic aspects of curves. When presented with any curve, a natural task is to compute the number of points on it with coordinates in some finite field. When the finite field is large this is generally difficult to do. Ren'e Schoof gave a polynomial time algorithm for counting points on elliptic curves i.e., those of genus 1, in his ground...
The modpn library: Bringing fast polynomial arithmetic into maple
 IN MICA’08
, 2008
"... We investigate the integration of C implementation of fast arithmetic operations into Maple, focusing on triangular decomposition algorithms. We show substantial improvements over existing Maple implementations; our code also outperforms Magma on many examples. Profiling data show that data conversi ..."
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Cited by 17 (15 self)
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We investigate the integration of C implementation of fast arithmetic operations into Maple, focusing on triangular decomposition algorithms. We show substantial improvements over existing Maple implementations; our code also outperforms Magma on many examples. Profiling data show that data conversion can become a bottleneck for some algorithms, leaving room for further improvements.
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.
Testing sign conditions on a multivariate polynomial and applications
 Mathematics in Computer Science
"... Abstract. Let f be a polynomial in Q[X1,..., Xn] of degree D. We focus on testing the emptiness and computing at least one point in each connected component of the semialgebraic set defined by f> 0 (or f < 0 or f = 0). To this end, the problem is reduced to computing at least one point in each con ..."
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Cited by 14 (5 self)
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Abstract. Let f be a polynomial in Q[X1,..., Xn] of degree D. We focus on testing the emptiness and computing at least one point in each connected component of the semialgebraic set defined by f> 0 (or f < 0 or f = 0). To this end, the problem is reduced to computing at least one point in each connected component of a hypersurface defined by f − e = 0 for e ∈ Q positive and small enough. We provide an algorithm allowing us to determine a positive rational number e which is small enough in this sense. This is based on the efficient computation of the set of generalized critical values of the mapping f: y ∈ C n → f(y) ∈ C which is the union of the classical set K0(f) of critical values of the mapping f and K∞(f) of asymptotic critical values of the mapping f. Then, we show how to use the computation of generalized critical values in order to obtain an efficient algorithm deciding the emptiness of a semialgebraic set defined by a single inequality or a single inequation. At last, we show how to apply our contribution to determining if a hypersurface contains real regular points. We provide complexity estimates for probabilistic versions of the latter algorithms which are within O(n 7 D 4n) arithmetic operations in Q. The paper ends with practical experiments showing the efficiency of our approach.
A concise proof of the Kronecker polynomial system solver from scratch
, 2006
"... Abstract. Nowadays polynomial system solvers are involved in sophisticated computations in algebraic geometry as well as in practical engineering. The most popular algorithms are based on Gröbner bases, resultants, Macaulay matrices, or triangular decompositions. In all these algorithms, multivariat ..."
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Cited by 9 (1 self)
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Abstract. Nowadays polynomial system solvers are involved in sophisticated computations in algebraic geometry as well as in practical engineering. The most popular algorithms are based on Gröbner bases, resultants, Macaulay matrices, or triangular decompositions. In all these algorithms, multivariate polynomials are expanded in a monomial basis, and the computations mainly reduce to linear algebra. The major drawback of these techniques is the exponential explosion of the size of the polynomials needed to represent highly positive dimensional solution sets. Alternatively, the “Kronecker solver ” uses data structures to represent the input polynomials as the functions that compute their values at any given point. In this paper we present the first selfcontained and student friendly version of the Kronecker solver, with a substantially simplified proof of correctness. In addition, we enhance the solver in order to compute the multiplicities of the zeros without any extra cost.
DEFORMATION TECHNIQUES FOR SPARSE SYSTEMS
, 2006
"... Abstract. We exhibit a probabilistic symbolic algorithm for solving zero– dimensional sparse systems. Our algorithm combines a symbolic homotopy procedure, based on a flat deformation of a certain morphism of affine varieties, with the polyhedral deformation of Huber and Sturmfels. The complexity of ..."
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Cited by 4 (1 self)
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Abstract. We exhibit a probabilistic symbolic algorithm for solving zero– dimensional sparse systems. Our algorithm combines a symbolic homotopy procedure, based on a flat deformation of a certain morphism of affine varieties, with the polyhedral deformation of Huber and Sturmfels. The complexity of our algorithm is quadratic in the size of the combinatorial structure of the input system. This size is mainly represented by the mixed volume of Newton polytopes of the input polynomials and an arithmetic analogue of the mixed volume associated to the deformations under consideration. 1.
Highperformance symbolic computation in a hybrid compiledinterpreted programming environment
 In ICCSA’08
, 2008
"... We investigate the integration of C implementation of fast arithmetic operations into MAPLE, focusing on triangular decomposition algorithms. We show substantial improvements over existing MAPLE implementations; our code also outperforms MAGMA on many examples. Profiling data show that data conversi ..."
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Cited by 4 (4 self)
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We investigate the integration of C implementation of fast arithmetic operations into MAPLE, focusing on triangular decomposition algorithms. We show substantial improvements over existing MAPLE implementations; our code also outperforms MAGMA on many examples. Profiling data show that data conversion can become a bottleneck for some algorithms, leaving room for further improvements. 1
A baby steps/giant steps probabilistic algorithm for computing roadmaps in smooth bounded real hypersurface
, 2009
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