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I'm trying to draw the Riemann surface defined by the polynomial $z^2=w^5-w$, where $z$ and $w$ are complex numbers. I'm a new user of Mathematica, and I tried manipulating some examples I found, but with no luck. Could someone help me, please?

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  • $\begingroup$ Have you seen this answer? $\endgroup$
    – C. E.
    Feb 19, 2017 at 9:32
  • $\begingroup$ Yes I have seen it, but I don't know how to get to work. I'll try again. $\endgroup$
    – yaa09d
    Feb 19, 2017 at 22:16
  • $\begingroup$ I'm supposed to get a genus 2 surface but I don't get that! $\endgroup$
    – yaa09d
    Feb 19, 2017 at 23:04
  • $\begingroup$ Take a look at: mathematica-journal.com/2010/02/… $\endgroup$
    – Stratus
    Sep 17, 2017 at 15:24

2 Answers 2

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Trott gives a simple technique in his article on rendering Riemann surfaces. At the time, he had said:

Although in principle this method works (…) for higher order polynomials the Resultant calculation will become much more expensive. Also, the generation of a high resolution ContourPlot3D requires much CPU time…

Nowadays, with more powerful computers, the method is now slightly more feasible to do:

expr = Expand[w^5 - w - z^2 /. {z -> x + I y, w -> u + I v}];

ip = Total[Expand[Cases[expr, _Complex _]/I]];
rp = Expand[expr - I ip];

resRe = Resultant[rp, ip, v];
resIm = Resultant[rp, ip, u];

{ContourPlot3D[resRe == 0, {x, -7, 7}, {y, -7, 7}, {u, -2, 2}, 
               AxesLabel -> {"x", "y", "u"}, BoundaryStyle -> None, 
               BoxRatios -> {3, 3, 2}, ContourStyle -> Opacity[2/3, ColorData[97, 3]],
               MaxRecursion -> 1, Mesh -> False, PlotLabel -> "real part",
               PlotPoints -> 55, PlotRange -> All],
 ContourPlot3D[resIm == 0, {x, -7, 7}, {y, -7, 7}, {v, -2, 2}, 
               AxesLabel -> {"x", "y", "v"}, BoundaryStyle -> None, 
               BoxRatios -> {3, 3, 2}, ContourStyle -> Opacity[2/3, ColorData[97, 1]],
               MaxRecursion -> 1, Mesh -> False, PlotLabel -> "imaginary part",
               PlotPoints -> 55, PlotRange -> All]} // GraphicsRow

Riemann surface for z^2 = w^5 - w

The artifacts along the surfaces' self-intersections is due to my inability to increase PlotPoints and MaxRecursion on my computer; a better computer than mine may be able to make a better picture.

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ResourceFunction["RiemannSurfacePlot3D"][z^2 == w^5 - w, Re[z], {w, z}]

enter image description here

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