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Say $\{a_1,a_2,..,a_n\}$ and $\{b_1,b_2,...,b_n\}$ be the real roots of two monic polynomials of degree $n$ which have a common interlacing. (say I have arranged the roots in increasing order)

  • Can one estimate for how large a positive integer $k$ is it possible that, $\sum_i a_i ^k = \sum_i b_i^k$ might hold?

  • And given a positive integer say $p$ is there a good lower bound on the term, $\sum_i (a_i^p - b_i^p)$ ?

DEFINITION :

These two polynomials will be said to have a "common interlacing" if there exits numbers $\{ c_1,..,c_{n-1}\}$ such that $c_i \in [a_i ,a_{i+1}] \cap [b_i, b_{i+1}]$. (this is basically saying that it is possible to construct another polynomial whose every root is always simultaneously between two consecutive roots of both the polynomials)

A simple example of two polynomials which are not of "common interlacing" is to consider $(x-1)(x-3)$ and $(x-4)(x-5)$.

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  • $\begingroup$ What does "have a common interlacing" mean? $\endgroup$
    – Hugh Thomas
    Commented Jun 27, 2015 at 23:07
  • $\begingroup$ Edited the question to add in the definition. Hope you can help! $\endgroup$
    – Student
    Commented Jun 29, 2015 at 2:00
  • $\begingroup$ Thanks. Now I am not sure I understand the quantifiers. For suitable $a_i$ and $b_i$, you could have the equation hold for arbitrarily large k: just make $a_1 << 0$ and $b_n >>0$, and then tune their values to make the equation hold (while fixing the other values). $\endgroup$
    – Hugh Thomas
    Commented Jun 29, 2015 at 12:58
  • $\begingroup$ (Oops, in my previous comment, I meant to assume $k$ is even, so that $a_1^k$ and $b_n^k$ will predominate.) $\endgroup$
    – Hugh Thomas
    Commented Jun 29, 2015 at 16:36

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