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
Continuations and monodromy on random riemann surfaces
Proceedings of the 2009 conference on Symbolic numeric 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
Roots of the derivatives of some random polynomials
Proceedings of the 2011 International Workshop on Symbolic-Numeric Computation
Hi-index | 5.23 |
We consider a Riemann surface X defined by a polynomial f(x,y) of degree d, whose coefficients are chosen randomly. Hence, we can suppose that X is smooth, that the discriminant @d(x) of f has d(d-1) simple roots, @D, and that @d(0)0, i.e. the corresponding fiber has d distinct points {y"1,...,y"d}. When we lift a loop 0@?@c@?C-@D by a continuation method, we get d paths in X connecting {y"1,...,y"d}, 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 loops around each point of @D. Multiplying families of ''neighbor'' transpositions, we construct permutations and the subgroups of the symmetric group they generate. This allows us to establish and study experimentally two conjectures on the distribution of these transpositions and on transitivity of the generated subgroups. Assuming that these two conjectures are true, we develop tools allowing fast probabilistic algorithms for absolute multivariate polynomial factorization, under the hypothesis that the factors behave like random polynomials whose coefficients follow uniform distributions.