Let $f$ be a real valued continuous algebraic function on $\mathbb R^n$. Suppose the zero set of $f$ is bounded, i.e., if $|x|$ is large enough, $f(x)\neq 0$. Is there any estimate of the sort $|f(x)|\ge c(1+|x|)^{-N}$ for some $c>0$ and $n\in\mathbb N$? A prototypical example of such an inequality is $\sqrt{1+x^2}-x\ge c(1+|x|)^{-1}$, but I would appreciate general estimates along this line that apply to algebraic functions of $n$ variables.



Decay of real continuous algebraic functions at infinity
Your function $f$ is, in particular, a continuous semi-algebraic function on $\mathbf R^n$, i.e., its graph is a semi-algebraic subset of $\mathbf R^{n+1}$. Such functions are known to have sub-polynomial growth. More precisely (cf. Bochnak, Coste, Roy: Real algebraic geometry, Proposition 2.6.2, p. 43):


Let $S\subseteq\mathbf R^n$ be a closed semi-algebraic subset, and let $f\colon S\rightarrow \mathbf R$ be a continuous semi-algebraic function. Then there are $c\in\mathbf R$ and $N\in\mathbf N$ such that
$$
|f(x)|\leq c(1+||x||^2)^N
$$
for all $x\in S$.


Apply this to the inverse $1/f$ of your function on the closed semi-algebraic subset $||x||\geq A$, where $A\in\mathbf R$ is such that all zeros of $f$ have norm $<A$.


