This paper introduces a new efficient algorithm for computing Gröbner bases. To avoid as much as possible intermediate computation, the algorithm computes successive truncated Gröbner bases and it replaces the classical polynomial reduction found in the Buchberger algorithm by the simultaneous reduction of several polynomials. This powerful reduction mechanism is achieved by means of a symbolic precomputation and by extensive use of sparse linear algebra methods. Current techniques in linear algebra used in Computer Algebra are reviewed together with other methods coming from the numerical field. Some previously untractable problems (Cyclic 9) are presented as well as an empirical comparison of a first implementation of this algorithm with other well known programs. This comparison pays careful attention to methodology issues. All the benchmarks and CPU times used in this paper are frequently updated and available on a Web page. Even though the new algorithm does not improve the worst case complexity it is several times faster than previous implementations both for integers and modulo computations. 1

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

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.

We present a new algorithm to compute the Integer Smith normal form of large sparse matrices. We reduce the computation of the Smith form to independent, and therefore parallel, computations modulo powers of word-size primes. Consequently, the algorithm does not suffer from coefficient growth. We have implemented several variants of this algorithm (Elimination and/or Black-Box techniques) since practical performance depends strongly on the memory available. Our method has proven useful in algebraic topology for the computation of the homology of some large simplicial complexes.

ABSTRACT. A 3-manifold is Haken if it contains a topologically essential surface. The Virtual Haken Conjecture says that every irreducible 3-manifold with infinite fundamental group has a finite cover which is Haken. Here, we discuss two interrelated topics concerning this conjecture. First, we describe computer experiments which give strong evidence that the Virtual Haken Conjecture is true for hyperbolic 3-manifolds. We took the complete Hodgson-Weeks census of 10,986 small-volume closed hyperbolic 3-manifolds, and for each of them found finite covers which are Haken. There are interesting and unexplained patterns in the data which may lead to a better understanding of this problem. Second, we discuss a method for transferring the virtual Haken property under Dehn filling. In particular, we show that if a 3-manifold with torus boundary has a Seifert fibered Dehn filling with hyperbolic base orbifold, then most of the Dehn filled manifolds are virtually Haken. We use this to show that every non-trivial Dehn surgery on the figure-8 knot is virtually Haken.

A probabilistic algorithm is presented to find the determinant of a nonsingular, integer matrix. For a matrix A ¡£ ¢ n ¤ n the algorithm requires O ¥ n 3 ¦ 5 ¥ logn § 4 ¦ 5 § bit operations (assuming for now that entries in A have constant size) using standard matrix and integer arithmetic. Using asymptotically fast matrix arithmetic, a variant is described which requires O ¥ n 2 ¨ θ © 2 � log 2 nloglogn § bit operations, where two n � n matrices can be multiplied with O ¥ n θ § operations. The determinant is found by computing the Smith form of the integer matrix, an extremely useful canonical form in itself. Our algorithm is probabilistic of the Monte Carlo type. That is, it assumes a source of random bits and on any invocation of the algorithm there is a small probability of error. 1

In this paper we consider deterministic computation of the exact determinant of a dense matrix M of integers. We present a new algorithm with worst case complexity O \Gamma n 4 (log n + log jjM jj) + n 3 log 2 jjM jj \Delta , where n is the dimension of the matrix and jjM jj is a bound on the entries in M , but with average expected complexity O \Gamma n 4 + n 3 (log n + log jjM jj) 2 \Delta , assuming some plausible properties about the distribution of M . We will also describe a practical version of the algorithm and include timing data to compare this algorithm with existing ones. Our result does not depend on "fast" integer or matrix techniques.

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.

A simple randomized algorithm is given for finding an integer solution to a system of linear Diophantine equations. Given as input a system which admits an integer solution, the algorithm can be used to find such a solution with probability at least 1/2. The running time (number of bit operations) is essentially cubic in the dimension of the system. The analogous result is presented for linear systems over the ring of polynomials with coefficients from a field. 1 Introduction Solving a system of linear Diophantine equations is a classical mathematical problem: given an integer matrix A and vector b, the goal is to find an integer vector x that satisfies Ax = b. We present a simple randomized algorithm for solving this problem. We also show how to adapt the algorithm to solve linear systems over the ring of univariate polynomials with coefficients from a field. The algorithm is easy to implement, memory efficient, and faster than previous methods. The algorithm is probabilistic in th...

Abstract. A d-dimensional framework is a graph and a map from its vertices to E d. Such a framework is globally rigid if it is the only framework in E d with the same graph and edge lengths, up to rigid motions. For which underlying graphs is a generic framework globally rigid? We answer this question by proving a conjecture by Connelly, that his sufficient condition is also necessary: a generic framework is globally rigid if and only if it has a stress matrix with kernel of dimension d + 1, the minimum possible. An alternate version of the condition comes from considering the geometry of the lengthsquared mapping ℓ: the graph is generically locally rigid iff the rank of ℓ is maximal, and it is generically globally rigid iff the rank of the Gauss map on the image of ℓ is maximal. We also show that this condition is efficiently checkable with a randomized algorithm, and prove that if a graph is not generically globally rigid then it is flexible one dimension higher. 1.