Suppose I know the following information about a function :


1) Its a polynomial (not an explicit equation, neither the roots nor the degree is known)


2) I have managed to find an algebraic relation between some of the roots (mind you I do not know the roots explicitly, just the form of the algebraic relation is known to me).


Now given this information can one say something about the polynomial itself ?


Now what do I seek for? Well, information on something like the divisors of the degree of the polynomial, or say something about the Galois group of the polynomial may be ....
so you can say am asking an inverse question.


I understand that under these very general condition the problem may not even be well posed. I actually have more information about the polynomial in the particular case I encountered it ... the polynomial is a 0-1 polynomial ...some of the roots lie in the unit circle... etc. etc.


But certainly there would be instances of similar problems (with more information
available about the polynomial/ the nature and number of algebraic relations that are available etc.) which has been dealt with ?


So, I wanted to ask the question in a more general setting. Any variant of this I would say is quite interesting. So you can assume different kind of condition on the roots, coefficient algebraic relation,


I will greatly appreciate if some one can point out where I should be looking. Reference to literature where such a problem has been dealt with would be great.


Regards


Vagabond



A question regarding polynomials whose roots satisfy certain algebraic relation
This is rather vague but regarding relations between roots of a polynomial you may try some of Chris Smyth's papers as a starting point. For instance, this one:


C. J. Smyth, Conjugate algebraic numbers on conics, Acta Arith. 40 (1982), 333–346.


