A general condition on polynomial roots

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Discussion Overview

The discussion centers around the conditions on the coefficients of a polynomial to determine whether its roots can be real or possess even imaginary parts. The inquiry involves theoretical aspects of polynomial roots, potentially linking to concepts in Galois theory.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant proposes examining the polynomial's derivatives to identify turning points, suggesting that if turning points are on the same side of the x-axis, it indicates the presence of imaginary roots.
  • Another participant questions the setup of the polynomial, seeking clarification on the roles of the coefficients and the variable k, and how roots are defined in relation to k.
  • There is a reiteration of the need for clarification regarding the definitions of the variables and the nature of the roots, particularly concerning the relationship between w(k) and w(-k).

Areas of Agreement / Disagreement

Participants express confusion regarding the initial setup and definitions, indicating a lack of consensus on the interpretation of the polynomial and its roots.

Contextual Notes

There are unresolved questions about the definitions of the coefficients and the variable k, as well as the implications of the polynomial's structure on the nature of its roots.

lunogled
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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
 
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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.
 
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...
 
morphism said:
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
 

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