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 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 quasi-linear complexity. The main outcome we have in mind is the acceleration of higher-level algorithms, by interfacing our low-level implementation with languages such as AXIOM or Maple. We show the potential for huge speed-ups, by comparing two AXIOM implementations of van Hoeij and Monagan's modular GCD algorithm.

In [25] and [22] a new algorithmic concept was introduced for the symbolic solution of a zero dimensional complete intersection polynomial equation system satisfying a certain generic smoothness condition. The main innovative point of this algorithmic concept consists in the introduction of a new geometric invariant, called the degree of the input system, and the proof that the most common elimination problems have time complexity which is polynomial in this degree and the length of the input.

New algorithms are presented for factoring polynomials of degree n over the finite field of q elements, where q is a power of a fixed prime number. When log q = n 1+a , where a ? 0 is constant, these algorithms are asymptotically faster than previous known algorithms, the fastest of which required time \Omega\Gamma n(log q) 2 ), y or \Omega\Gamma n 3+2a ) in this case, which corresponds to the cost of computing x q modulo an n degree polynomial. The new algorithms factor an arbitrary polynomial in time O(n 3+a+o(1) +n 2:69+1:69a ). All measures are in fixed precision operations, that is in bit complexity. Moreover, in the special case where all the irreducible factors have the same degree, the new algorithms run in time O(n 2:69+1:69a ). In particular, one may test a polynomial for irreducibility in O(n 2:69+1:69a ) bit operations. These results generalize to the case where q = p k , where p is a small prime number relative to q. 1 Introduction The expected run...

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.

We propose fast algorithms for computing composed products and composed sums, as well as diamond products of univariate polynomials. These operations correspond to special resultants, that we compute using power sums of roots of polynomials, by means of their generating series.

We study the complexity of arithmetic in finite fields of characteristic two, F2n. We concentrate on the following two problems: • Iterated Multiplication: Given α1, α2,...,αt ∈ F2 n, compute α1 · α2 · · ·αt ∈ F2 n. • Exponentiation: Given α ∈ F2 n and a t-bit integer k, compute αk ∈ F2 n.

We prove lower bounds of order n log n for both the problem of multiplying polynomials of degree n, and of dividing polynomials with remainder, in the model of bounded coefficient arithmetic circuits over the complex numbers. These lower bounds are optimal up to order of magnitude. The proof uses a recent idea of R. Raz [Proc. 34th STOC 2002] proposed for matrix multiplication. It reduces the linear problem of multiplying a random circulant matrix with a vector to the bilinear problem of cyclic convolution. We treat the arising linear problem by extending J. Morgenstern’s bound [J. ACM 20, pp. 305-306, 1973] in a unitarily invariant way. This establishes a new lower bound on the bounded coefficient complexity of linear forms in terms of the singular values of the corresponding matrix. In addition, we extend these lower bounds for linear and bilinear maps to a model of circuits that allows a restricted number of unbounded scalar multiplications.

Abstract. We report efficient implementation techniques for FFT-based dense multivariate polynomial arithmetic over finite fields, targeting multi-cores. We have extended a preliminary study dedicated to polynomial multiplication and obtained a complete set of efficient parallel routines in Cilk++ for polynomial arithmetic such as normal form computation. Since bivariate multiplication applied to balanced data is a good kernel for these routines, we provide an in-depth study on the performance and the cut-off criteria of our different implementations for this operation. We also show that, not only optimized parallel multiplication can improve the performance of higher-level algorithms such as normal form computation but also this composition is necessary for parallel normal form computation to reach peak performance on a variety of problems that we have tested.

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