On the roots of Bernoulli polynomials












4












$begingroup$


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?






reference-request cv.complex-variables soft-question polynomials







share|cite|improve this question









$endgroup$








  • 5




    $begingroup$
    One reference is arxiv.org/pdf/math/0703452.pdf.
    $endgroup$
    – Richard Stanley
    Jan 23 at 3:44










  • $begingroup$
    @RichardStanley: thank you much for the quick reply with the reference.
    $endgroup$
    – T. Amdeberhan
    Jan 23 at 4:43
















4












$begingroup$


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?






reference-request cv.complex-variables soft-question polynomials







share|cite|improve this question









$endgroup$








  • 5




    $begingroup$
    One reference is arxiv.org/pdf/math/0703452.pdf.
    $endgroup$
    – Richard Stanley
    Jan 23 at 3:44










  • $begingroup$
    @RichardStanley: thank you much for the quick reply with the reference.
    $endgroup$
    – T. Amdeberhan
    Jan 23 at 4:43














4












4








4


4



$begingroup$


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?






reference-request cv.complex-variables soft-question polynomials







share|cite|improve this question









$endgroup$




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?






reference-request cv.complex-variables soft-question polynomials




reference-request cv.complex-variables soft-question polynomials






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Jan 23 at 3:15









T. AmdeberhanT. Amdeberhan

17.2k229127




17.2k229127








  • 5




    $begingroup$
    One reference is arxiv.org/pdf/math/0703452.pdf.
    $endgroup$
    – Richard Stanley
    Jan 23 at 3:44










  • $begingroup$
    @RichardStanley: thank you much for the quick reply with the reference.
    $endgroup$
    – T. Amdeberhan
    Jan 23 at 4:43














  • 5




    $begingroup$
    One reference is arxiv.org/pdf/math/0703452.pdf.
    $endgroup$
    – Richard Stanley
    Jan 23 at 3:44










  • $begingroup$
    @RichardStanley: thank you much for the quick reply with the reference.
    $endgroup$
    – T. Amdeberhan
    Jan 23 at 4:43








5




5




$begingroup$
One reference is arxiv.org/pdf/math/0703452.pdf.
$endgroup$
– Richard Stanley
Jan 23 at 3:44




$begingroup$
One reference is arxiv.org/pdf/math/0703452.pdf.
$endgroup$
– Richard Stanley
Jan 23 at 3:44












$begingroup$
@RichardStanley: thank you much for the quick reply with the reference.
$endgroup$
– T. Amdeberhan
Jan 23 at 4:43




$begingroup$
@RichardStanley: thank you much for the quick reply with the reference.
$endgroup$
– T. Amdeberhan
Jan 23 at 4:43










1 Answer
1






active

oldest

votes


















12












$begingroup$

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






share|cite|improve this answer











$endgroup$













  • $begingroup$
    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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1 Answer
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active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









12












$begingroup$

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






share|cite|improve this answer











$endgroup$













  • $begingroup$
    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
















12












$begingroup$

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






share|cite|improve this answer











$endgroup$













  • $begingroup$
    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














12












12








12





$begingroup$

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






share|cite|improve this answer











$endgroup$



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







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Jan 27 at 3:14

























answered Jan 23 at 3:54









Robert IsraelRobert Israel

41.9k51120




41.9k51120












  • $begingroup$
    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


















  • $begingroup$
    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
















$begingroup$
Many thanks for the quick and generous response to my request for the plots.
$endgroup$
– T. Amdeberhan
Jan 23 at 4:44




$begingroup$
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




$begingroup$
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




$begingroup$
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




$begingroup$
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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