Viewing polynomials as functions

Question

I would like to show, from first principles, that $n$-variables complex polynomials can be identified with a subset of the set of all functions $\mathbb{C}^n\to\mathbb{C}$. In other words, that the natural map from the ring of polynomials $\mathbb{C}[x_1,\ldots,x_n]$ to the ring of all functions $\mathbb{C}^n\to\mathbb{C}$ given by evaluation of polynomials is an injective ring homomorphism.

Thus, we must show that if $$f(x_1,\ldots,x_n)=\sum_{I}a_{I}x^I\in\mathbb{C}[x_1,\ldots,x_n]$$ and $f(x_1,\ldots,x_n)=0$ for all $(x_1,\ldots,x_n)\in\mathbb{C}^n$ then $a_I=0$ for all index sets $I$. In other words, we must show that the monomials $x^I$ are linearly independent in $\mathbb{C}[x_1,\ldots,x_n]$. Is their a simple elementary proof of that well-known fact?

Answer 1

Prove the contrapositive by induction on $n$.

For $n=0$ it's trivial.

If it's true for $n-1$, write $f(x_1, \ldots, x_n) = \sum_{j=0}^m g_j(x_1,...,x_{n-1}) x_n^j$ where $g_j$ are polynomials in $n-1$ variables, at least one of which is not $0$ (as a polynomial). Let $g_k$ be the one of these with lowest index. By the induction hypothesis, there is some $(a_1,\ldots,a_{n-1}) \in \mathbb C^{n-1}$ with $g_k(a_1, \ldots,a_{n-1}) = t \ne 0$. Then $$ \lim_{x \to 0} \frac{f(a_1, \ldots, a_{n-1}, x)}{x^k} = g_k(a_1, \ldots, a_k) = t$$
so $f(a_1,\ldots, a_{n-1},x)$ can't always be $0$.