Numerical Computation of Galois Groups

Abstract

The Galois/monodromy group of a family of geometric problems or equations is a subtle invariant that encodes the structure of the solutions. Computing monodromy permutations using numerical algebraic geometry gives information about the group, but can only determine it when it is the full symmetric group. We give numerical methods to compute the Galois group and study it when it is not the full symmetric group. One algorithm computes generators while the other gives information on its structure as a permutation group. We illustrate these algorithms with examples using a Macaulay2 package we are developing that relies upon Bertini to perform monodromy computations.

Algebraic Curves A ∘ l ( x ) − U ( y ) = 0 and Arithmetic of Orbits of Rational Functions

“…Proof. The proof is based on the following geometric description of G X and S X (see, e.g., [5], §I.G or [7], Section 2.2). Let L be the n-fold fiber product of X : C → CP 1 with itself, that is, the algebraic curve in C n defined by the equation…”

Lower bounds for genera of fiber products

“…§I.G, or [29], Section 2.2). For k, 1 ď k ď n, let L k,V be an algebraic variety consisting of k-tuples of E k a common image under V , and p L k,V a variety obtained from L k,V by removing points that belong to the big diagonal ∆ k,E :" tpx i q P E k | x i " x j for some i ‰ ju of E k .…”

Real monodromy action

“…In the one parameter case, the monodromy group is generated by the permutations arising from the finitely many loops that generate the fundamental group of the intersection of U to the line parameterized by ptq. This is described in detail with numerical algebraic geometric computations in [8] and illustrated in the following example.…”

“…3.15. Moreover, the (complex) monodromy group computed using [8] is S 6 , which is not a solvable group.…”

“…The complex numbers bestow many properties on the monodromy group such as it is base point independent and does not change when restricting to a general curve section of the parameter space [17]. These simplify the computation of the monodromy group [8] summarized in § 2.…”