Consider a non-constant multivariate real analytic function $f$ on $\mathbb{R}^n$. My question is, can the zeros of $f$ be dense in $\mathbb{R}^n$? In one dimension, I know that they cannot be, as the zeros of a single variable non-constant real analytic function of a single variable cannot have an accumulation point. Any help is appreciated, thanks!
2025-01-13 00:08:25.1736726905
Zeros of real analytic functions on $\mathbb{R}^n$
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By assumption, the function is continuous. The zero set of continuous functions is always closed, as it is the pre-image of $\{0\}$.
The closure of a dense set is the full domain. Per assumption the zero set of your function is dense and closed, thus the full domain. The only function satisfying your conditions is the zero function.