Continuations and monodromy on random riemann surfaces

Abstract

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,...,y_d}. When we lift a loop 0 ∈ γ ⊂ C - Δ by a continuation method, we get d paths in X connecting {y₁,...,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 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.