Suppose $f(x)=\sum\_{|\alpha|=0}^{\infty}a\_{\alpha} x^{\alpha}$ for all $x\in\mathbb{R}^n$. Moreover we know a priori that $f$ is an algebraic function.
Is $f$ necessarily a polynomial?If not what are typical counterexamples?



Smooth algebraic functions
You wrote a series. If you mean that it converges for all $x\in R^n$ than this is an entire
function that is the series also converges for $x\in C^n$. An entire algebraic function is
a polynomial, indeed. This follows from the growth estimate and Cauchy's inequalities, for example.


