Say that a polynomial in an indeterminate $x$ with real coefficients of degree $d$ has positive coefficients if each of the coefficients of $x^d,\ldots,x^1,x^0$ is (strictly) positive.

For $f$ a monic univariate polynomial with real coefficients, if there exists a positive integer $m_0$ such that its $m_0$th power $f^{m_0}$ has positive coefficients, then does there exist a positive integer $m_1$ such that for each positive integer $m$ at least $m_1$, the $m$th power $f^m$ has positive coefficients?

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    $\begingroup$ $f(x)=-x-1$, $m_0=2$? $\endgroup$ – Emil Jeřábek Sep 9 '14 at 14:56
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    $\begingroup$ I think one should rule out that sort of counterexample by saying the polynomial is monic. $\endgroup$ – Douglas Zare Sep 9 '14 at 15:03
  • $\begingroup$ ah, thanks Emil and Douglas. I will at the monic condition in. $\endgroup$ – user2529 Sep 9 '14 at 15:26

Yes. For nonnegative (as opposed to strictly positive) coefficients, a more general result is available (in several variables). See the article by me on powers of polynomials in Symbolic Dynamics and Its Applications (Editors, Adler and Walters, AMS 1992 (unfortunately, I can't find a link to it) [I have three articles there]. In several variables, the formulation is, if $P$ is a polynomial such that $P(1,1,\dots,1)> 0$, and there exists $m$ such that $P^m$ has only nonnegative coefficients, then for all sufficiently large $M$, $P^M$ has only nonnegative coefficients.

In one variable, the strict positivity condition you require can also be dealt with from this, but is a nonstarter in more than two variables. Note that if the second largest or the second smallest degree monomial has zero coefficient, then that persists in all powers, so no power of such poly in one variable can be strictly positive in your sense. However, the answer to your question is yes, because the gaps will eventually fill in ... (this requires a little, but not much, work).

Edit: Here is the reference (found, finally, in Zentralblatt): Handelman, David, Polynomials with a positive power. (Zbl 0794.26013) Symbolic dynamics and its applications, Proc. AMS Conf. in honor of R. L. Adler, New Haven/CT (USA) 1991, Contemp. Math. 135, 229-230 (1992). [For the entire collection see Zbl 0755.00019.]

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Yes. This is a part of my answer to another MO question,

Zeros of polynomials with real positive coefficients,

@David Handelman: thanks for the reference! I will insert it to my paper.

EDIT. In fact, Theorem 1 in the preprint cited above is not new. This was found as a result of David's answer. The revised preprint is posted on the arxiv and here http://www.math.purdue.edu/~eremenko/dvi/saddle13.pdf

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  • $\begingroup$ After reading the answers by you and Prof Handleman and having a discussion with my PhD supervisor, we realized that the affirmative answer to my question and in fact what is now Theorem A in your revised preprint is a corollary of an isometric embedding theorem of Catlin and D'Angelo, MRL 1999. There, Catlin and D'Angelo proved, in particular, that for a globalizable Hermitian metric on a holomorphic line bundle over a compact complex manifold, if this metric is negatively curved and satisfies the so called Sharp Global Cauchy Schwarz inequality, then sufficiently large powers of this metric $\endgroup$ – user2529 Sep 18 '14 at 13:34
  • $\begingroup$ can be isometrically embedded into complex projective space. Theorem A can be deduced from this result of Catlin and D'Angelo in the case where the complex compact manifold is the Riemann sphere and using the fact that every negative holomorphic line bundle over the Riemann surface is antiample. $\endgroup$ – user2529 Sep 18 '14 at 13:37
  • $\begingroup$ In particular, as you noted, the condition that $|f(z)|<f(|z|)$ is equivalent to the associated Hermitian metric of $f$ satisfying the Sharp Global Cauchy Schwar inequality. The other condition that the first two and the last two coefficients of $f$ are strictly positive is equivalent to its associated globalizable Hermitian metric being negatively curved. $\endgroup$ – user2529 Sep 18 '14 at 13:56
  • $\begingroup$ @Colin Tan: we knew about Catlin and d'Angelo results, and even discussed "our" theorem A with each of them. We could not derive Theorem A from their statements. I mean the relation of the condition about "first two and last two" with their conditions. $\endgroup$ – Alexandre Eremenko Sep 19 '14 at 4:38

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