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algcurves

 monodromy
 Compute the monodromy of an algebraic curve

 Calling Sequence monodromy(f, x, y, opt)

Parameters

 f - irreducible polynomial in x and y x - variable y - variable opt - optional arguments

Description

 • 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 $N=\mathrm{degree}\left(f,y\right)$ is the degree of covering. Curves with singularities are allowed as input.
 • The output is a list containing the following:
 1 A value $\mathrm{x0}$ for x for which y takes N different values, so that $\mathrm{x0}$ is not a branchpoint nor a singularity.
 2 A list $L=\left[\mathrm{fsolve}\left(\mathrm{subs}\left(x=\mathrm{x0},f\right),y,\mathrm{complex}\right)\right]$ of pre-images of $\mathrm{x0}$. This list of y-values at $x=\mathrm{x0}$ effectively labels the sheets of the Riemann surface at $x=\mathrm{x0}$. Sheet 1 is ${L}_{1}$, sheet 2 is ${L}_{2}$, and so on.
 3 A list$\left[\left[{b}_{1},{m}_{1}\right],\left[{b}_{2},{m}_{2}\right],\mathrm{...}\right]$ of branchpoints ${b}_{i}$ with their monodromy ${m}_{i}$. The monodromy ${m}_{i}$ of branchpoint ${b}_{i}$ is the permutation of $L$ obtained by applying analytic continuation on $L$ following a path from $\mathrm{x0}$ to ${b}_{i}$, going around ${b}_{i}$ counter-clockwise, and returning to $\mathrm{x0}$.
 • The permutations ${m}_{i}$ will be given in disjoint cycle notation. The branchpoints ${b}_{i}$ are roots of $\mathrm{discrim}\left(f,y\right)$.
 • The order of the branchpoints is chosen in such a way that the complex numbers ${b}_{1}-\mathrm{x0},...$ have increasing arguments. The point x0 is chosen on the left of the branchpoints, so all arguments are between $-\frac{\mathrm{\pi }}{2}$ and $\frac{\mathrm{\pi }}{2}$. 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 ${m}_{i}$.  This group G is the Galois group of f as a polynomial over $C\left(x\right)$. G is a subgroup of galois(f,y), which is the Galois group of f over Q(x).

Examples

 > $\mathrm{with}\left(\mathrm{algcurves}\right):$
 > $\mathrm{monodromy}\left({y}^{3}-x,x,y\right)$
 $\left[{-1.}{,}\left[{-1.}{,}{0.500000000000000}{-}{0.866025403784439}{}{I}{,}{0.500000000000000}{+}{0.866025403784439}{}{I}\right]{,}\left[\left[{0.}{,}\left[\left[{1}{,}{2}{,}{3}\right]\right]\right]{,}\left[{\mathrm{\infty }}{,}\left[\left[{1}{,}{3}{,}{2}\right]\right]\right]\right]\right]$ (1)
 > $f≔{\left({y}^{4}-x\right)}^{2}+1$
 ${f}{≔}{\left({{y}}^{{4}}{-}{x}\right)}^{{2}}{+}{1}$ (2)
 > $G≔\mathrm{monodromy}\left(f,x,y,\mathrm{group}\right)$
 ${G}{≔}{\mathrm{permgroup}}{}\left({8}{,}\left\{\left[\left[{1}{,}{5}{,}{8}{,}{4}\right]\right]{,}\left[\left[{2}{,}{3}{,}{7}{,}{6}\right]\right]{,}\left[\left[{1}{,}{4}{,}{8}{,}{5}\right]{,}\left[{2}{,}{6}{,}{7}{,}{3}\right]\right]\right\}\right)$ (3)

Note: G is not transitive, which means that f is reducible.

 > $\mathrm{evala}\left(\mathrm{AFactor}\left(f\right)\right)$
 $\left({-}{{y}}^{{4}}{+}{x}{-}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{1}\right)\right){}\left({-}{{y}}^{{4}}{+}{x}{+}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{1}\right)\right)$ (4)