Partition a disk into regions based on points on a circle

Here is another way to implement the same thing as JasonB. I'm assuming that sols is the roots given by your code.

angles = Sort@N[ArcTan @@@ sols];
cones = Partition[First[angles] + Accumulate@Differences[angles], 2, 1];

cones is a list of intervals of angles representing the cones. Two intervals are missing due to how the list is computed, we add them manually:

AppendTo[cones, {2 Pi + First[angles], Last[angles]}];
AppendTo[cones, angles[[1 ;; 2]]];

Now we can visualize the slices if we want to:

disks = Transpose[{
    Array[ColorData[97], Length[cones]],
    Disk[{0, 0}, 1, #] & /@ cones
    }];

Graphics[{
  disks,
  Red, PointSize[Medium], Point[sols]
  }]

This is not ideal but in case similar procedure for other values of k. A modification of function. Once segment is not colored.

dr[p_, r_, k_] := 
 Module[{}, sols = Solve[(x^2)^(1/r) + (y^2)^(1/k) == 1 && p == 0];
  sols = {x, y} /. sols;
  g1 = ListPlot[sols, PlotStyle -> {Red}, 
    PlotMarkers -> {Automatic, 10}];
  g2 = ContourPlot[(x^2)^(1/r) + (y^2)^(1/k) == 1, {x, -1, 1}, {y, -1,
      1}];
  reg = DiscretizeRegion@
    ImplicitRegion[(x^2)^(1/r) + (y^2)^(1/k) < 
      1, {{x, -1, 1}, {y, -1, 1}}];
  sg = DiscretizeRegion[
      Disk[{0, 0}, 
       Max[Norm@#1, Norm@#2], {ArcTan @@ #1, ArcTan @@ #2}]] & @@@ 
    Partition[SortBy[N@sols, Pi/2 + ArcTan @@ # &], 2, 1];
  int = RegionIntersection[reg, #] & /@ sg;
  Show[g1, g2, ##, AspectRatio -> Automatic, Frame -> True, 
     PlotLabel -> Row[{"k= ", k}], 
     ImageSize -> 
      250] & @@ (RegionPlot[#, 
       PlotStyle -> {RandomColor[], Opacity[0.5]}] & /@ int)]

Example:

Grid[Partition[
  dr[84 x^7*y + 380 x^6*y^2 + 509 x^5*y^3 - 509 x^3*y^5 - 
      380 x^2*y^6 - 84 x*y^7, 1, #] & /@ Range[9], 3]]

This function will take any points and create what is essentially a pie chart from them,

pointsPieChart[pts_, radius_: 1, center_: {0, 0}] := 
 Module[{angles},
  angles = 
   ArcTan @@@ pts // Sort // 
    Append[#, First@# + 2 π] &;
  Graphics[
   Table[{ColorData[97][n], 
     Disk[center, radius, angles[[n ;; n + 1]]]}, {n, Length@pts}]]
  ]

It works by

• First arranging the points in order around a circle (by finding the convex hull of the points, extracting the polygon from and using the points therein, a trick described here. Thanks to C.E. for showing that this step isn't necessary, since the angles are being sorted anyway.

• Then the angles for these points are found using ArcTan, appending one final angle onto the end (equal to the original angle plus 2 π)

• Then using Disk to get the segment of the disk between two angles, and combine them all into one Graphics object.

Here is the function using random points from a unit circle, and I show the points just to verify that the function is working:

Table[
 pts = RandomPoint[Circle[], 9];
 Show[
  pointsPieChart[pts],
  Graphics[{Red, PointSize[Medium], Point@pts}]], {10}]

and here it is applied to the points in the OP,

pointsPieChart[sols]