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I have generated the following plot:

enter image description here

Code:

RegionPlot3D[  x^2 + z^2 < 1 &&   x^2 + y^2 < 1, {x, -2, 2}, {y, -2, 2}, {z, -2, 2}, Axes -> False]

However, the resolution is a bit poor in the plot above. How can I increase the polygon count or increase the resolution some other way?

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Grid[{RegionPlot3D[
     x^2 + z^2 < 1 && x^2 + y^2 < 1, {x, -2, 2}, {y, -2, 2}, {z, -2, 2}, Axes -> False, 
     Mesh -> False, PlotPoints -> #, ViewPoint -> {2, 0, 2}, Boxed -> False] & /@ 
                                                                         {Automatic, 10, 20, 100}}]

Mathematica graphics

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Just change PlotPoints. Here values between 10 and 100:

enter image description here

Creation code:

gr = Table[
   Rasterize@
    RegionPlot3D[
     x^2 + z^2 < 1 && x^2 + y^2 < 1, {x, -2, 2}, {y, -2, 2}, {z, -2, 
      2}, Axes -> False, PlotPoints -> i, Mesh -> All], {i, 10, 100, 
    5}];

Export["tmp/resolution.gif", Join[gr, Reverse[gr]], 
 "DisplayDurations" -> .3]
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