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  1. PO
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    <p>Ended up trying myself since the goal really was to do it 'hands off'. I'll leave the question open for a good while to see if anybody finds a better way.</p> <p>The code below uses bisection to bracket the points where <code>CountRoots</code> changes value. This works for my case (spotting the singularity at x=0 is pure luck):</p> <pre><code>In[214]:= findRootBranches[Function[x, Evaluate@geyvals[[1, 1]]], {-5, 5}] Out[214]= {{{-5., -0.0158768}, 1}, {{-0.0158768, -5.96046*10^-9}, 3}, {{0., 0.}, 2}, {{5.96046*10^-9, 1.05635}, 3}, {{1.05635, 5.}, 1}} </code></pre> <p>Implementation:</p> <pre><code>Options[findRootBranches] = { AccuracyGoal -&gt; $MachinePrecision/2, "SamplePoints" -&gt; 100}; findRootBranches::usage = "findRootBranches[f,{x0,x1}]: Find the the points in [x0,x1] \ where the number of real roots of a polynomial changes. Returns list of {&lt;interval&gt;,&lt;root count&gt;} pairs. f: Real -&gt; Polynomial as pure function, e.g f=Function[x,#^2-x&amp;]." ; findRootBranches[f_, {xa_, xb_}, OptionsPattern[]] := Module[ {bisect, y, rootCount, acc = 10^-OptionValue[AccuracyGoal]}, rootCount[x_] := {x, CountRoots[f[x][y], y]}; (* Define a ecursive bisector w/ automatic subdivision *) bisect[{{x1_, n1_}, {x2_, n2_}} /; Abs[x1 - x2] &gt; acc] := Module[{x3, n3}, {x3, n3} = rootCount[(x1 + x2)/2]; Which[ n1 == n3, bisect[{{x3, n3}, {x2, n2}}], n2 == n3, bisect[{{x1, n1}, {x3, n3}}], True, {bisect[{{x1, n1}, {x3, n3}}], bisect[{{x3, n3}, {x2, n2}}]}]]; (* Find initial brackets and bisect *) Module[{xn, samplepoints, brackets}, samplepoints = N@With[{sp = OptionValue["SamplePoints"]}, If[NumberQ[sp], xa + (xb - xa) Range[0, sp]/sp, Union[{xa, xb}, sp]]]; (* Start by counting roots at initial sample points *) xn = rootCount /@ samplepoints; (* Then, identify and refine the brackets *) brackets = Flatten[bisect /@ Cases[Partition[xn, 2, 1], {{_, a_}, {_, b_}} /; a != b]]; (* Reinclude the endpoints and partition into same-rootcount segments: *) With[{allpts = Join[{First@xn}, Flatten[brackets /. bisect -&gt; List, 2], {Last@xn}]}, {#1, Last[#2]} &amp; @@@ Transpose /@ Partition[allpts, 2] ]]] </code></pre>
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    1. POMathematica: branch points for real roots of polynomial
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    1. CO-1: This doesn't spot the cases where two roots cross each other.
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    2. COOk, so finding the precise location of a root crossing is very doable, since Mathematica can do series expansions of root objects: `x/.First@Solve[0==Normal@Quiet@Series[r1-r2,{x,x0,1}],x]`. Now, how to spot potential crossings is not so obvious...
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