Expressing a polynomial in $n$ variables in the form of a summation

I have been working on the following problem:

Let $A$ be a ring containing a field $F$. If $a_1 ,\dots, a_n \in A$, show that there is a unique ring homomorphism $\varphi : F[x_1,\dots,x_n] \to A$ with $\varphi(x_i)=a_i$ for each $i$.

I have already thought enough about this question and I do not know how to express a polynomial which has $n$ variables in a summation. One knows that if it is a polynomial in $x$, then $f(x)=\sum_{i=0}^{n} a_i x^i$, but if $f(x_1,\dots,x_n)$ is a polynomial in $n$ variables then how can I express it in the form of a summation?

• You can use multi-index notation. For each $\alpha = (\alpha_1, ... , \alpha_n) \in \mathbb{N}^n$, define $$X^{\alpha} = X_1^{\alpha_1} \cdots X_n^{\alpha_n}$$ Then every element of $F[X_1, ... , X_n]$ can be expressed uniquely as $$f(X_1, ... , X_n) = \sum\limits_{\alpha} c_{\alpha}X^{\alpha}$$ where $c_{\alpha}$ is an element of $F$ which is $0$ for all but finitely many $\alpha$. – D_S Sep 2 '17 at 22:54
• Another thing you can do is induction: you can identify the rings $F[X_1, ... , X_{n-1}][X_n]$. So by the one variable case, every element of $F[X_1, ... , X_n]$ can be expressed uniquely as $$\sum\limits_{k=0}^m f_k X^k$$ where each $f_k$ is a polynomial in $n-1$ variables. – D_S Sep 2 '17 at 22:59
• What about $\varphi(\sum_\alpha c_\alpha X^\alpha) = \sum_\alpha \sigma(c_\alpha) A^\alpha$ for some $\sigma \in Aut(F)$ or $Aut(A/F)$ ? What condition on $A$ can you find so that $\varphi$ is an homomorphism iff $\sigma$ is the identity ? – reuns Sep 3 '17 at 0:55

You can do it iteratively: $$\sum_{k_1=0}^{m_1}\cdots\sum_{k_n=0}^{m_n}b_{k_1}^1\cdots b_{k_n}^{m}x_1^{k_1}\cdots x_n^{k_n}$$ Or you can list the terms by degree (which is useful in some scenarios): $$\sum_{k=0}^{m}\left(\sum_{k_1 + \cdots + k_n = k}b_{k_1}^1\cdots b_{k_n}^{m}x_1^{k_1}\cdots x_n^{k_n}\right)$$