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The documentation for FindInstance[] says:

The instances returned by FindInstance typically correspond to special or interesting points in the set.

Nothing further is given, and a search of the phrase is likewise fruitless. Does anybody have any insight onto what exactly they mean by this, or maybe some specific examples that might illustrate the idea?

Edit: I suppose some insight into the algorithm used would provide a partial answer to the question.

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Here is a counter-example from ?FindInstance to that statement. Let us consider

FindInstance[ x^2 + y^2 + z^2 <= 1 && 9 z^3 == 2 x - 5 y - 7, {x, y, z}, Reals]

{{x->45/128,y->-(1/2),z->-(3/4)}}

and draw it by

a = RegionPlot3D[ x^2 + y^2 + z^2 <= 1,{x, -1, 1},{y, -1, 1}, {z, -1, 1},PlotStyle -> Opacity[0.5]];
b = Graphics3D[{PointSize[Large], Red, Point[{45/128, -(1/2), -(3/4)}]}];
c = ContourPlot3D[{9 z^3 == 2 x - 5 y - 7}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}];Show[{a, b,c}]

enter image description here

We see nothing special/interesting for Point[{45/128, -(1/2), -(3/4)}].

PS. The same conclusion for the results of FindInstance[ x^2 + y^2 + z^2 <= 1 && 9 z^3 == 2 x - 5 y - 7, {x, y, z}, Reals,5].

PPS. FindInstance[x^2 + y^2 == z^2 && x > 0 && y > 0 && z > 0, {x, y, z}, Integers]

{{x -> 8, y -> 6, z -> 10}}

FindInstance[x^2 + y^2 == z^2, {x, y, z}, Integers, 3]

{{x -> 0, y -> 980, z -> 980}, {x -> 975156, y -> -3254045, z -> 3397019}, {x -> -2952, y -> -26486, z -> -26650}}

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  • $\begingroup$ Interesting - so it's definitely not always an 'interesting point', whatever that means. For a specific example I'm using, which is too complicated to make reproducing it here worthwhile, the result was very close to a singular point which got me thinking about this. $\endgroup$ – dbx Apr 9 '19 at 20:21
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    $\begingroup$ Try this: FindInstance[x > 5, x], then FindInstance[x > 5, x, 2], then FindInstance[x > 5, x, 3]. Notice how none of the solutions from these overlap... so when looking for multiple answers, the definition of "interesting" must change. That seems interesting. Or this: FindInstance[x > #, x] & /@ Range[10] you can see that what is interesting is always 27 greater than the argument. $\endgroup$ – bill s Apr 10 '19 at 3:41
  • $\begingroup$ @bill s: Sorry, don't understand it. I will be waiting for serious comments. $\endgroup$ – user64494 Apr 10 '19 at 4:22

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