[seqfan] Re: Number of real roots of polynomial with coefficients +-1, or 0, +-1
W. Edwin Clark
wclark at mail.usf.edu
Thu Feb 23 23:19:13 CET 2017
Concerning A282692 <https://oeis.org/A282691>, it appears that one counts
the multiplicity of the roots.
You have a(7) = 3, however, Maple tells me that
x^7+x^6-x^5-x^4-x^3-x^2+x+1 = (x^2+1)*(-1+x)^2*(1+x)^3
and so the roots are 1,1,-1,-1,-1. I get a(7) = 5.
Otherwise my calculations agree. It is interesting that the irreducible
for the polynomials with coefficients 1, -1 for degree n have the same form
for n < 16 with the exception of n = 7, 11 and 15. Probably the strong law
of small numbers
and the fact that most polynomials over Z are irreducible.
Here's my Maple program for the computation of a(n) using Strum's Theorem
square free factorization of the 1,-1 polynomials.
while not T[finished] do
for u in q do k:=k+numroots(u,x)*u; od;
if k > m then m:=k; P:=p; fi;
a:=proc(n) b(n); end proc:
On Thu, Feb 23, 2017 at 1:34 PM, Neil Sloane <njasloane at gmail.com> wrote:
> https://oeis.org/A282691, A282692, A282693 all need more terms!
> Seqfan Mailing list - http://list.seqfan.eu/
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