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Consider the Bernoulli polynomials denoted by $B_n(z)$. Now, start plotting the set of all (combined) complex roots $\mathcal{A}_N$ of $B_n(z)$, say for $n=1,2,\dots,N$ for some enough large $N$. It appears that $\mathcal{A}_N$ branches into several "curves".

QUESTION: Or, does it? If so, what are these curves?

Request: Can someone post the complex plot here?

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Here is an animation of the zeros of the first $100$ Bernoulli polynomials, produced using Maple.

enter image description here

For the number of real roots, see OEIS sequence A094937 and references there.

EDIT: As requested by Wolfgang, here is a plot of the real roots for even $n$ up to $200$.

enter image description here

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  • $\begingroup$ Many thanks for the quick and generous response to my request for the plots. $\endgroup$ – T. Amdeberhan Jan 23 at 4:44
  • $\begingroup$ That is nice! It looks like the real roots essentially oscillate between even and odd degrees. Could you add a plot only for, say, even n? $\endgroup$ – Wolfgang Jan 26 at 15:31
  • $\begingroup$ Very nice! Can share the code, please? Put it here or link to github or like that... $\endgroup$ – Alexander Chervov Jan 27 at 14:41
  • $\begingroup$ plots[display]([seq(plots[pointplot](map([Re,Im],[fsolve(bernoulli(n,z),z,complex)])),n=1..100)],insequence=true); plots[pointplot]([seq(op(map(t -> [n,t],[fsolve(bernoulli(n,z),z)])),n=2..200,2)],symbol=point,axes=box); $\endgroup$ – Robert Israel Jan 27 at 16:45

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