FAQFAQ   SearchSearch   MemberlistMemberlist   UsergroupsUsergroups 
 ProfileProfile   PreferencesPreferences   Log in to check your private messagesLog in to check your private messages   Log inLog in 
Forum index » Science and Technology » Math » Recreational
Poly. degree four = ?
Post new topic   Reply to topic Page 1 of 1 [6 Posts] View previous topic :: View next topic
Author Message
Alex. Lupas
science forum Guru Wannabe


Joined: 06 May 2005
Posts: 245

PostPosted: Tue Jun 07, 2005 3:51 am    Post subject: Poly. degree four = ? Reply with 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 ]
Back to top
Virgil
science forum Guru


Joined: 24 Mar 2005
Posts: 5536

PostPosted: Tue Jun 07, 2005 5:48 am    Post subject: Re: Poly. degree four = ? Reply with quote

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 ?
Back to top
Timothy Little
science forum Guru Wannabe


Joined: 30 May 2005
Posts: 295

PostPosted: Tue Jun 07, 2005 6:10 am    Post subject: Re: Poly. degree four = ? Reply with quote

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
Back to top
Alex. Lupas
science forum Guru Wannabe


Joined: 06 May 2005
Posts: 245

PostPosted: Tue Jun 07, 2005 12:02 pm    Post subject: Re: Poly. degree four = ? Reply with quote

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


Joined: 01 May 2005
Posts: 412

PostPosted: Tue Jun 07, 2005 3:46 pm    Post subject: Re: Poly. degree four = ? Reply with quote

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
coefficients, so your problem simplifies to asking what pairs of
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
Back to top
Alex. Lupas
science forum Guru Wannabe


Joined: 06 May 2005
Posts: 245

PostPosted: Wed Jun 08, 2005 12:15 am    Post subject: Re: Poly. degree four = ? Reply with quote

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<v), such that r^u=r^v

that is
======
r^s=1
======
where s:=v-u is a positive integer. In conclusion each

element from Q must be a root of unity, i.e. in our case

there exists positive integers s_1,s_2 ;k_1,k_2,1=<k_j=<s_j,

such that


z_1=cos(2k_1*pi/s_1) + i*sin(2k_1*pi/s_1)

z_2=cos(2k_1*pi/s_1) - i*sin(2k_1*pi/s_1)

z_3=cos(2k_2*pi/s_2) + i*sin(2k_2*pi/s_2)

z_4=cos(2k_2*pi/s_2) - i*sin(2k_2*pi/s_2)

.. ......
Back to top
Google

Back to top
Display posts from previous:   
Post new topic   Reply to topic Page 1 of 1 [6 Posts] View previous topic :: View next topic
The time now is Fri Nov 24, 2017 9:04 am | All times are GMT
Forum index » Science and Technology » Math » Recreational
Jump to:  

Similar Topics
Topic Author Forum Replies Last Post
No new posts How to break this USA heat wave of 104 degree F; solution... a_plutonium@hotmail.com Chem 7 Mon Jul 17, 2006 7:31 pm
No new posts Student Factor and degree of freedom dh Prediction 2 Thu Jun 29, 2006 3:17 pm
No new posts Fifth Degree Equation bassam king karzeddin Math 19 Wed May 31, 2006 3:17 pm
No new posts " When is f(1,y) ^ [n] a p degree poynomial? " alain verghote Math 3 Mon May 22, 2006 10:23 am
No new posts Ring with nilpotent elements of arbitrary degree Seán O'Leathlóbhair Math 11 Thu May 18, 2006 11:06 am

Copyright © 2004-2005 DeniX Solutions SRL
Other DeniX Solutions sites: Electronics forum |  Medicine forum |  Unix/Linux blog |  Unix/Linux documentation |  Unix/Linux forums  |  send newsletters
 


Powered by phpBB © 2001, 2005 phpBB Group
[ Time: 0.0473s ][ Queries: 16 (0.0320s) ][ GZIP on - Debug on ]