Let $ f \in k[x_1, \ldots,x_n]$, $k$ a field then it seems to me that $f$ has only finitely many roots in $k^n$. I was trying induction but did not really work:
$n=1$ follows from Euclidean algorithm. Suppose holds $ \le n-1$, write polynomial with coefficients in $k[x_1]$.For each fixed choice of $x_1 = a \in k$, there are finitely many solutions…
Any hints?
Best Answer
This statement is false in general. Consider the following polynomials in $\mathbb{C}[X,Y]$
The polynomials $ XY $ and $ X + Y $ have infinitely many zeros in $ \mathbb{C}^{2} $.
The zero-set of $XY$ is union of $(\{0\}×\mathbb{C})$ and $(\mathbb{C}×\{0\})$, while the zero-set of $X+Y$ is $\{(a,−a) | a\in \mathbb{C}\}$.
Both are uncountable sets!!