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Let $P(x)=1+a_1x+a_2x^2+a_3x^3+...$ be a series such that every $a_i$ is an integer, $a_1<0$, and $a_i\ge 0$ for every $i\ge 2$. Are the following statements equivalent ?

  • $P(y)=0$ for some $y>0$.
  • Every coefficient of the series expansion of $\frac 1{P(x)}$ is positive.
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Yes. One direction is quite clear, the other one is carefully written in

D. I. Piotkovskii, "On the growth of graded algebras with a small number of defining relations", Uspekhi Mat. Nauk, 48:3(291) (1993), 199–200.

(It is also mentioned without proof in Lemma 5.3 of D. Anick, "Generic algebras and CW complexes", Algebraic Topology and Algebraic K-Theory (Proc. Conf. Princeton, NJ (USA)), Ann. Math. Stud., vol. 113 (1987), pp. 247–321)

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