Suppose $f \in \mathbb C[X_1,\ldots,X_n]$ is a complex polynomial in $n$ variables such that $f(P) = 0$ for all $P \in \mathbb R^n$. Is then necessarily $f = 0$?
This is certainly true for $n=1$ as a univariate polynomial that vanishes in infinitely many points is the zero polynomial. However, this argument does not immediately generalize to the case $n > 1$, e.g. $f(X,Y) = X-Y$ has infinitely many zeroes but is not the zero polynomial.