Large Galois groups with applications to Zariski density

Abstract

We introduce the first provably efficient algorithm to check if a finitely generated subgroup of an almost simple semi-simple group over the rationals is Zariski-dense. We reduce this question to one of computing Galois groups, and to this end we describe efficient algorithms to check if the Galois group of a polynomial p with integer coefficients is "generic" (which, for arbitrary polynomials of degree n means the full symmetric group S n , while for reciprocal polynomials of degree 2n it means the hyperoctahedral group C 2 ≀S n .). We give efficient algorithms to verify that a polynomial has Galois group S n , and that a reciprocal polynomial has Galois group C 2 ≀S n . We show how these algorithms give efficient algorithms to check if a set of matrices G in SL(n, Z) or Sp(2n, Z) generate a Zariski dense subgroup. The complexity of doing this inSL(n, Z) is of order O(n 4 log n log G ) log ǫ and in Sp(2n, Z) the complexity is of order O(n 8 log n log G ) log ǫ In general semisimple groups we show that Zariski density can be confirmed or denied in time of order O(n 1 4 log G log ǫ), where ǫ is the probability of a wrong "NO" answer, while G is the measure of complexity of the input (the maximum of the Frobenius norms of the generating matrices). The algorithms work essentially without change over algebraic number fields, and in other semi-simple groups. However, we restrict to the case of the special linear and symplectic groups and rational coefficient in the interest of clarity.

Four random permutations conjugated by an adversary generate Sn with high probability

“…It is a result going back to van der Waerden [vdW34] that most polynomials p(x) ∈ Z[x] of degree n have Galois group S n . Computing the Galois group is a central problem in computational number theory and is a fundamental building block for the solution of seemingly unrelated problems (see [Riv13] for an extensive discussion). Therefore, one cannot take for granted being in the "generic" case and one would like an effective and speedy algorithm for determining whether the Galois group of p(x) is the full symmetric group.…”

“…There are deterministic polynomial time algorithms to answer this. The first is due to S. Landau; a simpler and more efficient algorithm was proposed by the third author (see [Riv13]). These algorithms, however, are of purely theoretical interest due to their very long run times (their complexity is of the order of O(n 40 ), where n is the degree of the polynomial).…”

An origami of genus 3 with arithmetic Kontsevich–Zorich monodromy

“…[MMY,§6.7]). By the Zariski density criterion of Prasad-Rapinchuk [PR,Theorem 9.10] (see also [Ri,Theorem 1.5]), we have that ρ(Aff(O 1 )) is Zariski-dense in Sp(H…”

“…Anyhow, once we detect a good candidate origami O, the first step is the computation of its Kontsevich-Zorich monodromy, i.e., the Zariski closure of ρ(Aff(O)) (compare with Subsection 4.1). Here, the criterion of Prasad-Rapinchuk [PR,Theorem 9.10] (see also [Ri,Theorem 1.5]) informally says that the Zariski closure is "often" a symplectic group Sp or a product of SL 2 's. Moreover, the techniques in [MMY] indicate that the Zariski closure tends to be a symplectic group in many situations including H(4), but this must be taken with a grain of salt because the case of products of SL 2 happens in nature: for instance, Eskin-Kontsevich-Zorich [EKZ2] noted that the so-called "stairs" origamis in H(2g − 2) and H(g − 1, g − 1) are covered by special "square-tiled cyclic covers" and this information can be used to show that the Kontsevich-Zorich monodromy of a "stairs" origami is contained 13 in a product of SL 2 's.…”

Statistics of Random 3-Manifolds occasionally fibering over the circle

“…Remark 3.2. Whether or not a subgroup of a semisimple group is Zarsiki-dense can be efficiently determined using the results of [72].…”