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.



Conjectured equivalent conditions on certain power-series
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)


