Algebraic function $u\in C^\infty(\Bbb R^n)$

Question

Suppose $u\in C^{\infty}(\mathbb{R})$ is an algebraic function that has no singularities. Can it be said that $u$ is a polynomial?

If instead $u\in C^\infty(\Bbb R^n)$ and is an algebraic function. Is it also a polynomial?

Answer 1

too good to be true. Here is a middle school counterexample: Let $u(x)=\frac{1}{x^2+1}$ then $(x^2+1)u(x)-1=0$ for all $x\in\mathbb{R}.$