[seqfan] Re: Number of real roots of polynomial with coefficients +-1, or 0, +-1

[israel at math.ubc.ca]

A slight improvement: since (+/-)P((+/-)x) have the same number of real 
roots as P(x), you need only consider the cases where the x^0 and x^1 
coefficients are +1.

Cheers,
Robert

On Feb 23 2017, W. Edwin Clark wrote:

>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!
>>
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>>
>
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