A general condition on polynomial roots

[lunogled]

Consider a polynomial of the following type:

A_n w^n + A_{n-1} w^{n-1}k + A_{n-2} w^{n-2} k^2 + ... + A_1 k^n =0

What are the general conditions on {A_i} in order for the roots w(k) to be EITHER real OR functions with even imaginary parts, Im[w[k]]=Im[w[-k]]?

I would be interested in whether anyone has ever worked on this problem, or if this problem was ever shown to be unsolvable (perhaps there is a trivial connection to, e.g., Galois theory I missed).

Thanks, and best

 

[chiro]

Hey lunogled and welcome to the forums.

One way to look at this problem is in terms of the derivatives. If you order your turning points in order and you find that a pair of ordered points occurs on the same side of the x-axis, then you will have an imaginary root.

Think of it this way: if you have a real root, then the function will cross the x-axis at some point between the two-turning points and since one turning point will happen at a positive y value and another turning point will happen at a negative y value then if the function is continuous, it has to have a real root because it has to touch the x-axis and not only that, it will touch it only once due to some theorems in analysis/calculus but you can use an intuitive argument if you wish.

So easiest way to tell for a standard y = f(x) function, if you have this kind of condition, you should get an imaginary root if this happens, and to check this you need to solve the information for your first and possibly second derivatives (for inflection points) of your function.

 

[morphism]

I'm confused by your setup. Where do the A_n's live? Is w the indeterminate? If so, why do you have k's in there (i.e. why are you homogenizing)? If w(k) is a root, what does it mean for w(-k) to be a root? Is the k in w(k) a subscript or are you somehow viewing w as a function of k?

Please clarify...

 

[lunogled]

I'm confused by your setup. Where do the A_n's live? Is w the indeterminate? If so, why do you have k's in there (i.e. why are you homogenizing)? If w(k) is a root, what does it mean for w(-k) to be a root? Is the k in w(k) a subscript or are you somehow viewing w as a function of k?

Please clarify...

A_{n} are numbers. w as a function of k is indeterminate, so if w(k) is a root, w(-k) is also a root, w is defined as a function of k

 

[blue_raver22]

regards

The general conditions on the coefficients {A_i} for the roots w(k) to be either real or have even imaginary parts can be determined by considering the discriminant of the polynomial. The discriminant is a function of the coefficients and is equal to zero if and only if the polynomial has at least one multiple root. This means that in order for the roots to be either real or have even imaginary parts, the discriminant must be non-zero.

In terms of Galois theory, the problem of determining the conditions for the roots to be real or have even imaginary parts is equivalent to finding the Galois group of the polynomial. If the Galois group is solvable, then the roots can be expressed in terms of radicals and therefore can be either real or have even imaginary parts. However, if the Galois group is not solvable, then there may not be a solution in terms of radicals and the problem may be unsolvable.

It is possible that this problem has been studied in the context of Galois theory, but it is not a commonly studied problem in mathematics. Further research and investigation may be needed to determine if this problem has been previously solved or if it is still an open question.