compute the monodromy of an algebraic curve
monodromy(f, x, y, opt)
irreducible polynomial in x and y
This procedure computes the monodromy of a Riemann surface represented as a plane algebraic curve; that is, as a polynomial f in two variables x and y. The Riemann surface is the covering surface for y as an N-valued function of x, where N=degree⁡f,y is the degree of covering. Curves with singularities are allowed as input.
The output is a list containing the following:
A value x0 for x for which y takes N different values, so that x0 is not a branchpoint nor a singularity.
A list L=fsolve⁡subs⁡x=x0,f,y,complex of pre-images of x0. This list of y-values at x=x0 effectively labels the sheets of the Riemann surface at x=x0. Sheet 1 is L1, sheet 2 is L2, and so on.
A listb1,m1,b2,m2,... of branchpoints bi with their monodromy mi. The monodromy mi of branchpoint bi is the permutation of L obtained by applying analytic continuation on L following a path from x0 to bi, going around bi counter-clockwise, and returning to x0.
The permutations mi will be given in disjoint cycle notation. The branchpoints bi are roots of discrim⁡f,y.
The order of the branchpoints is chosen in such a way that the complex numbers b1−x0,... have increasing arguments. The point x0 is chosen on the left of the branchpoints, so all arguments are between −π2 and π2. If the arguments coincide, branchpoints that are closer to x0 are considered first. The point infinity will be given last, if it is a branchpoint.
It can take some time for this procedure to finish. To have monodromy print information about the status of the computation while it is working, give the variable infolevel[algcurves] an integer value > 1.
If the optional argument showpaths is given, then a plot is generated displaying the paths used for the analytic continuation. If the optional argument group is given, then the output is the monodromy group G, the permutation group generated by the mi. This group G is the Galois group of f as a polynomial over C⁡x. G is a subgroup of galois(f,y), which is the Galois group of f over Q(x).
f ≔ y4−x2+1
G ≔ monodromy⁡f,x,y,group
Note: G is not transitive, which means that f is reducible.
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