Computing monodromy via continuation methods on random Riemann surfaces

Abstract

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 δ(x) of f has d(d − 1) simple roots, ∆, and that δ(0) = 0 i.e. the corresponding fiber has d distinct points {y1,. .. , y d }. When we lift a loop 0 ∈ γ ⊂ C − ∆ 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 loops around each point of ∆. 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.

Roots of the derivatives of some random polynomials

“…It extends our previous works [12], [8], [10], [11]. Our conjectures, hopefully transformed into theorems, could then act as an oracle and indicate the estimated number and locations of the roots of a random polynomial and of its derivatives.…”

Solving polynomial systems via homotopy continuation and monodromy

“…We make the simplest possible assumption of uniform distribution on S d in order to perform the theoretical analysis in §4 and shed some light on why our framework works well. There is an interesting, more involved, alternative to this assumption in [12,11], which relies on the intuition in the case n = 1.…”