Factoring rational polynomials over the complex numbers
SIAM Journal on Computing
A numerical absolute primality test for bivariate polynomials
ISSAC '97 Proceedings of the 1997 international symposium on Symbolic and algebraic computation
Challenges of symbolic computation: my favorite open problems
Journal of Symbolic Computation
Towards factoring bivariate approximate polynomials
Proceedings of the 2001 international symposium on Symbolic and algebraic computation
Approximate multivariate polynomial factorization based on zero-sum relations
Proceedings of the 2001 international symposium on Symbolic and algebraic computation
Symmetric Functions Applied to Decomposing Solution Sets of Polynomial Systems
SIAM Journal on Numerical Analysis
Factoring multivariate polynomials via partial differential equations
Mathematics of Computation
Approximate factorization of multivariate polynomials via differential equations
ISSAC '04 Proceedings of the 2004 international symposium on Symbolic and algebraic computation
Lifting and recombination techniques for absolute factorization
Journal of Complexity
Approximate bivariate factorization: a geometric viewpoint
Proceedings of the 2007 international workshop on Symbolic-numeric computation
Computing monodromy groups defined by plane algebraic curves
Proceedings of the 2007 international workshop on Symbolic-numeric computation
Good reduction of puiseux series and complexity of the Newton-Puiseux algorithm over finite fields
Proceedings of the twenty-first international symposium on Symbolic and algebraic computation
From an approximate to an exact absolute polynomial factorization
Journal of Symbolic Computation
Good reduction of Puiseux series and applications
Journal of Symbolic Computation
Polynomial homotopies on multicore workstations
Proceedings of the 4th International Workshop on Parallel and Symbolic Computation
Computing monodromy via continuation methods on random Riemann surfaces
Theoretical Computer Science
Sampling algebraic sets in local intrinsic coordinates
Computers & Mathematics with Applications
A study of Hensel series in general case
Proceedings of the 2011 International Workshop on Symbolic-Numeric Computation
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Our main motivation is to analyze and improve factorization algorithms for bivariate polynomials in C[x,y], which proceed by continuation methods. We consider a Riemann surface X defined by a polynomial f(x,y) of degree d, whose coefficients are choosen randomly. Hence we can supose that X is smooth, that the discriminant δ(x) of f has d(d-1) simple roots, Δ, that δ(0) ≠ 0 i.e. the corresponding fiber has d distinct points {y1,...,yd}. When we lift a loop 0 ∈ γ ⊂ C - Δ by a continuation method, we get d paths in X connecting {y1,...,yd}, hence defining a permutation of that set. This is called monodromy. Here we present experimentations in Maple to get statistics on the distribution of transpositions corresponding to the loops turning around each point of Δ. Multiplying families of "consecutive" transpositions, we construct permutations then subgroups of the symmetric group. This allows us to establish and study experimentally some conjectures on the distribution of these transpositions then on transitivity of the generated subgroups. These results provide interesting insights on the structure of such Riemann surfaces (or their union) and eventually can be used to develop fast algorithms.