We briefly present and analyze, from a geometric viewpoint, strategies for designing algorithms to factor bivariate approximate polynomials in C[x, y]. Given a composite polynomial, stably square-free, satisfying a genericity hypothesis, we describe the effect of a perturbation on the roots of its discriminant with respect to one variable, and the perturbation of the corresponding monodromy action on a smooth fiber. A novel geometric approach is presented, based on guided projection in the parameter space and continuation method above randomly chosen loops, to reconstruct from a perturbed polynomial a nearby composite polynomial and its irreducible factors. An algorithm and its ingredients are described.

In this paper we design a stream cipher that uses the algebraic structure of the multiplicative group ZZ # p (where p is a big prime number used in ElGamal algorithm), by defining a quasigroup of order 1 and by doing quasigroup string transformations. The cryptographical strength of the proposed stream cipher is based on the fact that breaking it would be at least as hard as solving systems of multivariate polynomial equations modulo big prime number p which is NP-hard problem and there are no known fast randomized or deterministic algorithms for solving it. Unlikely the speed of known ciphers that work in ZZ # p for big prime numbers p, the speed of this stream cipher both in encryption and decryption phase is comparable with the fastest symmetrickey stream ciphers.

Ostrowski established in 1919 that an absolutely irreducible integral polynomial remains absolutely irreducible modulo all sufficiently large prime numbers. We obtain a new lower bound for the size of such primes in terms of the number of integral points in the Newton polytope of the polynomial, significantly improving previous estimates for sparse polynomials.

Abstract. In this paper, the problem of bounding the number of reducible curves in a pencil of algebraic plane curves is addressed. Unlike most of the previous related works, each reducible curve of the pencil is here counted with its appropriate multiplicity. It is proved that this number of reducible curves, counted with multiplicity, is bounded by d 2 − 1 where d is the degree of the pencil. Then, a sharper bound is given by taking into account the Newton’s polygon of the pencil.

Abstract. Indecomposable polynomials are a special class of absolutely irreducible polynomials. Some improvements of important effective results on absolute irreducibility have recently appeared using Ruppert’s matrix. In a similar way, we show in this paper that the use of a Jacobian matrix gives sharp bounds for the indecomposability problem. 1.

Abstract not found

Abstract. In this paper, the spectrum and the decomposability of a multivariate rational function are studied by means of the effective Noether’s irreducibility theorem given by Ruppert in [19]. With this approach, some new effective results are obtained. In particular, we show that the reduction modulo p of the spectrum of a given integer multivariate rational function r coincides with the spectrum of the reduction of r modulo p for p a prime integer greater or equal to an explicit bound. This bound is given in terms of the degree, the height and the number of variables of r. With the same strategy, we also study the decomposability of r modulo p. Some similar explicit results are also provided for the case of polynomials with coefficients in A = K[Z].

This paper presents an explicit bound on the number of primitive elements that are linear combinations of generators for field extensions.

In this paper we show how to compute the Galois group G of a polynomial f 2 Q(x)[Y ] by factoring the associated linear differential equation Lf (Y ) = 0 (and constructions of it) of minimal order satisfied by the roots of f . We use that the differential Galois group of Lf (Y ) is a faithful linear representation of G whose character is a summand of the permutation character of G acting on the roots of f . Our approach is motivated by the fact that the orders of the involved differential equations are much lower than the degrees of the Lagrange resolvants of f . In the final section we show how, if f 2 Q(x)[Y ], our approach via differential Galois theory helps one to also compute the Galois group of f over Q(x). 1. FROM POLYNOMIALS TO LINEAR DIFFERENTIAL EQUATIONS In the following we will consider the field Q, but the results remain valid for any field of characteristic 0. Let f = Y m + m\Gamma2 X i=0 b i (x)Y i 2 Q(x)[Y ] be a polynomial of degree m where the coefficient of Y m\Gamma1 is assumed to be zero 1 . We assume that f is an irreducible element of Q(x)[Y ] (i.e. that f is absolutely irreducible) and denote K the splitting field of f over Q(x) and z1 ; : : : ; zm the roots of f(Y ) = 0 in K. We now describe the well known construction of a linear differential equation with coefficients in Q(x) whose solution

Abstract. We study equations in groups G with unique m-th roots for each positive integer m. A word equation in two letters is an expression of the form w(X, A) = B, where w is a finite word in the alphabet {X, A}. We think of A, B ∈ G as fixed coefficients, and X ∈ G as the unknown. Certain word equations, such as XAXAX = B, have solutions in terms of radicals: X = A −1/2 (A 1/2 BA 1/2) 1/3 A −1/2 while others such as X 2 AX = B do not. We obtain the first known infinite families of word equations not solvable by radicals, and conjecture a complete classification. To a word w we associate a polynomial Pw ∈ Z[x, y] in two commuting variables, which factors whenever w is a composition of smaller words. We prove that if Pw(x 2, y 2) has an absolutely irreducible factor in Z[x, y], then the equation w(X, A) = B is not solvable in terms of radicals. 1.