A question regarding polynomials whose roots satisfy certain algebraic relation

Question

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

Answer 1

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.

Answer 2

I would guess that the most general way of going about this would be through the use of Gröbner bases, since you did mention that the root relations you have are algebraic in nature. To use a simple instance, here's how to reconstruct a quadratic polynomial in Mathematica from some simple algebraic relations of the roots a and b:

GroebnerBasis[{(x - a)(x - b), a + b - 5, a - 3b}, x, {a, b}]

(in English: find the (unique?) quadratic whose roots have a sum of 5 and has one root that is thrice the other)

which returns the polynomial $16x^2-80x+75$ (verify that the roots of this polynomial satisfy the preset conditions).

Of course, as with most algorithms of such generality, I would imagine this to be excruciatingly slow on a problem of even moderate complexity. Exploiting every little bit of structure present in your problem (which you say you have) would certainly help a great deal.

Answer 3

Such questions formed a substantial part of the classical "theory of equations", before Galois theory was formulated. For example, there is a book by Burnside on theory of equations, that is easy to find on Google Books. Whether you can get anything out of this, without knowing even the degree of the equation, is another matter. That suggests that you have some information about a field extension of unknown finite degree? Some theory about Tschirnhaus transformations might be of use to you, but I wonder who looks at those explicitly these days (maybe computer algebra people)? I get the impression of numerous special cases that are handled by direct algebra (tortuous exam questions, certainly).