How to define multi-variable polynomial using sequence?

Question

In his Basic Algebra, Knapp defines one-variable polynomials using sequences.

I am curious to find out the definition of multi-variable polynomials by using sequences.

Answer 1

Polynomial in $n$ indeterminates over a ring $R$ may be defined as the set of all multi-sequences $(a_{\nu})_{\nu\in\mathbb{N}}$ with $a_\nu\in R$ such that all but finitely many $a_\nu$ vanish.