algcurves
monodromy
compute the monodromy of an algebraic curve
Calling Sequence
Parameters
Description
Examples
monodromy(f, x, y, opt)
f
-
irreducible polynomial in x and y
x
variable
y
opt
optional arguments
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 is the degree of covering. Curves with singularities are allowed as input.
The output is a list containing the following:
A value for x for which y takes N different values, so that is not a branchpoint nor a singularity.
A list of pre-images of . This list of y-values at effectively labels the sheets of the Riemann surface at . Sheet 1 is , sheet 2 is , and so on.
A list of branchpoints with their monodromy . The monodromy of branchpoint is the permutation of obtained by applying analytic continuation on following a path from to , going around counter-clockwise, and returning to .
The permutations will be given in disjoint cycle notation. The branchpoints are roots of .
The order of the branchpoints is chosen in such a way that the complex numbers have increasing arguments. The point x0 is chosen on the left of the branchpoints, so all arguments are between and . 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 . This group G is the Galois group of f as a polynomial over . G is a subgroup of galois(f,y), which is the Galois group of f over Q(x).
Note: G is not transitive, which means that f is reducible.
See Also
algcurves[genus]
galois
Download Help Document