[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
factors
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
and the
square free factorization of the 1,-1 polynomials.

numroots:=proc(p,x)
local s:
sturm(sturmseq(p,x),x,-infinity,infinity):
end proc:

b:=proc(n)
local m,T,L,p,P,s,k,q,u;
m:=0;
T:=combinat:-cartprod([seq([1,-1],i=1..n+1)]):
while not T[finished] do
  L:=T[nextvalue]();
  p:=add(L[i]*x^(i-1),i=1..n+1);
  q:=sqrfree(p,x);
  k:=0;
  for u in q[2] do k:=k+numroots(u[1],x)*u[2]; od;
  if k > m then m:=k; P:=p; fi;
end do:
return [m,P];
end proc:

a:=proc(n) b(n)[1]; 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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