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Alex. Lupas
science forum Guru Wannabe

Joined: 06 May 2005
Posts: 245

Posted: Tue Jun 07, 2005 3:51 am    Post subject: Poly. degree four = ?

Suppose that A,B,C,D are integers and let f(x)=x^4+Ax^3+Bx^2+Cx+D
having the roots z_1,z_2,z_3,z_4 with |z_1|=|z_2|=|z_3|=|z_4|=1.
Find the coefficients A,B,C,D.
[Generalization !]

Possible source: [Leopold Kronecker, solution by Ludwig Bieberbach ]
Virgil
science forum Guru

Joined: 24 Mar 2005
Posts: 5536

Posted: Tue Jun 07, 2005 5:48 am    Post subject: Re: Poly. degree four = ?

In article <slrndaaejg.soe.tim-usenet@soprano.little-possums.net>,
Timothy Little <tim-usenet@little-possums.net> wrote:

 Quote: Alex. Lupas wrote: Suppose that A,B,C,D are integers and let f(x)=x^4+Ax^3+Bx^2+Cx+D having the roots z_1,z_2,z_3,z_4 with |z_1|=|z_2|=|z_3|=|z_4|=1. Find the coefficients A,B,C,D. Is there something wrong with the obvious A = B = C = 0, D = +-1? - Tim

Or even z_1 = z_2 = z_3 = z_4 = 1 so that A = C = -4, B = 6, D = 1 ?
Timothy Little
science forum Guru Wannabe

Joined: 30 May 2005
Posts: 295

Posted: Tue Jun 07, 2005 6:10 am    Post subject: Re: Poly. degree four = ?

Alex. Lupas wrote:
 Quote: Suppose that A,B,C,D are integers and let f(x)=x^4+Ax^3+Bx^2+Cx+D having the roots z_1,z_2,z_3,z_4 with |z_1|=|z_2|=|z_3|=|z_4|=1. Find the coefficients A,B,C,D.

Is there something wrong with the obvious A = B = C = 0, D = +-1?

- Tim
Alex. Lupas
science forum Guru Wannabe

Joined: 06 May 2005
Posts: 245

Posted: Tue Jun 07, 2005 12:02 pm    Post subject: Re: Poly. degree four = ?

Hi Timothy /Virgil,
I appreciate that your solution is'nt complete./Alex
Chip Eastham
science forum Guru

Joined: 01 May 2005
Posts: 412

Posted: Tue Jun 07, 2005 3:46 pm    Post subject: Re: Poly. degree four = ?

Alex. Lupas wrote:
 Quote: Suppose that A,B,C,D are integers and let f(x)=x^4+Ax^3+Bx^2+Cx+D having the roots z_1,z_2,z_3,z_4 with |z_1|=|z_2|=|z_3|=|z_4|=1. Find the coefficients A,B,C,D. [Generalization !] Possible source: [Leopold Kronecker, solution by Ludwig Bieberbach ]

Hi, Alex:

I notice that for roots on the unit circle, real implies +1,-1.

Removing such a factor preserves the integrality of polynomial
quadratic factors will multiply to give an integral quartic:

(x^2 + ax + b)(x^2 + cx + d)

with the constraint that the quadratics also have roots on the
unit circle.

But since the conjugate pairs of roots on the unit circle are
necessarily reciprocals, b = d = 1.

This is I believe enough to force a,c to be integers & polish
off the possibilities.

regards, chip
Alex. Lupas
science forum Guru Wannabe

Joined: 06 May 2005
Posts: 245

 Posted: Wed Jun 08, 2005 12:15 am    Post subject: Re: Poly. degree four = ? HINT:Denote by M the set of all f(x)=f(A,B,C,D;x)=x^4+Ax^3+Bx^2+Cx+D having the roots z_1,z_2,z_3,z_4 such that |z_1|=|z_2|=|z_3|=|z_4|=1. If Card(M) denotes the number of polynomials from M, then Card(M) is finite. This follows from the fact that |A|=|z_1+z_2+z_3+z_4| =< 4 |B|=|z_1z_2+z_1z_3+z_1z_4+z_2z_3+z_2z_4+z_3z_4| =< 6 |C| =< 6 |D| =< 1 . Let q be the number of all roots of polynomials f from M. Also q is a finite positive integer; denote by Q={r_1,r_2,....,r_q} the roots of all f, f in M. If r is in Q , then each numbers from r,r^2,r^3, ....,r^{q+1} is also in Q . Because Card(Q)=q there exists u,v, (u

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