# Difference between a polynomial of degree $n$ and an $n$- tuple with the $n$th component $\neq 0$?

When a polynomial $f := a_0 + a_1 X + \cdots + a_n X^n$ over a field with $a_n \neq 0$ cannot be regarded as a function, what is the major difference between $f$ and the $n$-tuple $(a_0,\dots, a_n)$? It seems that the plus sign and the indeterminate are redundant here.

• @Michael Hardy: What is the point of your "editing"?
– Yes
Commented Jul 10, 2015 at 3:14
• The important part is what algebraic structure they have, not so much how you represent them - polynomials are a very natural way to do so, but one may find analogous addition & multiplication laws for tuples. (Though still, setting $X$ as the analogous tuple, one retrieves the polynomial representation, because polynomials really are sums of powers of another element $X$) Commented Jul 10, 2015 at 3:32
• @Meelo : I changed $a_0 + a_1 X + \cdots a_n X^n$ to $a_0 + a_1 X + \cdots + a_n X^n$, which is standard usage and is better usage. Some people think that things like this matter only if they get noticed by someone. But in fact people often don't notice the things that cause them to mentally trip over something. ${}\qquad{}$ Commented Jul 10, 2015 at 4:39
• @MichaelHardy: You erroneously assumed something. Missing a plus sign is a typo, rather than you are more meticulous to mathematical writing. You generalize from one example to all...
– Yes
Commented Jul 10, 2015 at 4:54
• @Chou : Meelo asked what the point of my edit was. So I explained what it was. ${}\qquad{}$ Commented Jul 10, 2015 at 4:59

On the other hand, let me point out one advantage of using the classical notation $a_0 + a_1X + a_2X^2 + \cdots + a_nX^n$. The advantage of this is that it looks like a sum of "numbers", and sums of "numbers" is generally a "number". Since you can multiply "numbers", you should be able to multiply polynomials. Written in this classical form, it's easy to figure out how to multiply two polynomials - simply apply the distributive law multiple times. On the other hand, the tuple notation doesn't give any hint as to how one might "multiply" two polynomials. Do you multiply the corresponding entries in each tuple? What about multiplying two tuples of different lengths?
In fact none. The `formal' way of defining the polynomial ring over say a commutative ring $R$ is to identify it as the collection of all set functions $f:\mathbb{N}=\{0,1,2,\ldots\}\to R$ such that $f(n)=0$ for all large $n$. If $f,g$ are two such, we can define $(f+g)(n)=f(n)+g(n)$, $(fg)(n)=\sum_{i=0}^n f(i)g(n-i)$ and degree of $f$ is just the largest $n$ such that $f(n)\neq 0$.