Sparseness of real algebraic sets

Question

Let $f$ be a non-zero polynomial in $n$ variables with real coefficients. It seems intuitively clear that there are balls in $\mathbb{R}^n$ with arbitrarily large radius that do not meet the zero set of $f$. Is there a nice clean simple way to see this? If the question turns out to be hard, I'd be happy to see any proof at all.

Answer 1

Write $$ f(X_1,\ldots,X_n)=\sum_{e_1,\ldots, e_n\ge0}a_{e_1,\ldots,e_n}X_1^{e_1}\cdots X_n^{e_n}=\sum_{k\ge0}\underbrace{\sum_{e_1+\ldots +e_n=k}a_{e_1,\ldots,e_n}X_1^{e_1}\cdots X_n^{e_n}}_{=:g_k(X_1,\ldots,X_n)}$$ as sum of homogenuous polynomials and let $d>0$ me maximla with $g_d\ne0$. Then for all $X=(X_1,\ldots,X_n)$ we have $t^{-d}f(tX)\to g_d(X)$ as $t\to+\infty$. On the compact set $S^{n-1}$, this pointwise convergence of continous functions to a continuous limit is uniform. Since $g_d$ is not identically zero(!) and is homogenuous, it is not identically zero on $S^{n-1}$. Let $a\in S^{n-1}$ a point with $g(a)\ne0$. Then in some $r$-ball around $a$, $|g_d(x)|>\frac23|g_d(a)|$. For $t$ big enough, $|t^{-d}f(tx)-g_d(x)|<\frac13|g_d(a)|$ and hence $f(tx)\ne0$ whenever $t>t_0$, $x\in S^{n-1}$, $|x-a|<r$. The set of these $tx$ is an open cone minus a ball and contains arbitrarily large balls.