Assume that the formal power series $a(z)=\sum\limits\_{n\geq 1} a\_n z^n$ satisfies an algebraic equation with polynomial coefficients (that is, there exists a nonzero polynomial $F(z,y)$ such that $F(z,a(z))=0$), and a finite number of terms of $a(z)$ determine uniquely the whole series as a solution to that equation.


Assume also that the sequence $\{a\_n\}$ is increasing (and if necessary, one can even assume that all $a\_i$ are integers), and grows polynomially, that is for some $C,d$ we have $a\_n<Cn^d$ for all $n>>1$.


Is it true that there exists "a limit value of the exponent $d$"? More precisely, is it true that the infimum of exponents $d\_1$ such that $a\_n< C\_1 n^{d\_1}$ for some $C\_1$ (depending on $d\_1$) and all $n>>1$ coincides with the supremum of exponents $d\_2$ such that $a\_n > C\_2 n^{d\_2}$ for some $C\_2$ (depending on $d\_2$) and all $n>>1$?



Asymptotics/growth for coefficients of algebraic power series
This should follow from Theorem VII.8 in Flajolet and Sedgewick's [*Analytic Combinatorics*](http://algo.inria.fr/flajolet/Publications/books.html) (freely available at the link). As I understand the argument, you can obtain a straightforward general form for asymptotics of coefficients of algebraic generating functions by expanding in [Puiseux series](http://en.wikipedia.org/wiki/Puiseux_series) around the dominant poles. 


This implies the dominant term is a finite sum of terms of the form $n^d \zeta^n$ where $|\zeta| = 1$ for some $d \in \mathbb{Q}$, and the increasing growth condition should rule out the terms with $\zeta \neq 1$ messing up the asymptotics. 


