Why is polynomial defined only for n such that for each x^n term, n>= 0?
Basically, why not include more options for n? If we expand n to all integers (negative include), what is the problem? Likewise, can't we also allow n to be any real number?
Even so,it seems possible to make polynomial rings. Or am I missing something.
What is the reason that n is confined to only 0 and positive integers in the definition of polynomials?
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Let $R$ be a commutative ring. Then the free commutative ring generated from a set $\{X_1,\dots,X_n\}$ is the polynomial ring $R[X_1,\dots,X_n]$. In particular, univariate polynomials with coefficients in $R$ are the elements of the free commutative ring generated from a singleton set.
A key thing here is that a ring has only addition and multiplication. There is no exponentiation operation. A polynomial like $X^3+2$ "really" means $X\cdot X\cdot X + 2$ where $\cdot$ is the multiplication operation. The reason the exponents are restricted to natural numbers is $X^n$ really means $\underbrace{X\cdot\cdots\cdot X}_{n\text{ times}}$. It simply doesn't make sense to talk about a negative, rational, real, or complex "number" of times to multiple $X$ with itself.
In the Laurent "polynomial" case, which allows "negative" exponents, what we're really doing is making a bivariate polynomial ring $R[X,Y]$. We then quotient that ring by a congruence that makes $XY=1$ so the variable $Y$ behaves like $X^{-1}$. To a limited extent, we can do a similar thing for rational exponents, e.g. we could have $R[X,Y]/(Y^2-X)$ which would have $Y$ behave a bit like $X^{1/2}$.
When we talk about real or complex powers, we're usually working in a framework where we have an explicit exponential function which is not defined in relation to "repeated multiplication". For example, it may be a solution to the differential equation $Df=f$ and $f(0)=1$.
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The question "why" in this case is either meaningless or doesn't quite express your true quandary. I guess what you really want to know is along the lines of what is so special about the class of polynomials (in one indeterminate) that makes them so dominant -- right from school even into professional mathematics. Why can't we generalise them by using other numbers than positive integral exponents -- why, for example, not consider "polynomials" with negative integral exponents.
Well, indeed there is such a class of objects known as the rational functions (a ratio of two polynomials) and this generalises even further. I'm sure you've seen rational functions before but not in this light. For fractional exponents, one can define functions with them too -- but things. Indeed, one can define functions that differ from polynomials in that the exponents of the variable are real numbers. But the thing about all these "general forms" is that they do not produce the nice predictability of the polynomials. Consider, for example, that you can't really define a polynomial with real exponents, where the exponents range through all the values in a closed, bounded interval since intervals of the real line are uncountable.
In any case, the polynomials are a useful class of objects. In analysis they allow one to understand other functions because they are well understood (they're infinitely differentiable, integrable, etc.). They are just so useful that they suffice as the simplest examples in the study of other functions. This is the way the whole structure of mathematics is built -- more strange objects are understood only in light of the more familiar. For example, one cannot first teach real numbers to children before the natural numbers -- it's just wicked and wrong-headed. One first teaches them the positive integers and the positive rationals, which are easier to grasp and relate with. Only after that are they introduced to other number kinds. That they've not yet been so introduced might make one of the brighter ones among them think irrational numbers, say, have not been studied, but he would be so wrong, as I'm sure you know.
So, there are many different ways polynomials may be generalised (and there are those who study them) but polynomials are very basic and important in the study of the less familiar forms.
You can easily define "polynomials" with e.g. real exponents. Suppose $M$ is a commutative monoid, i.e. a set with an associative and commutative binary operation that has a unit element in $M$. Any abelian group is such a monoid, but also examples like the natural numbers $\mathbb N$ and the nonnegative real numbers $\mathbb R_{\geq 0}$. Now given any commutative ring $R$, you can construct the free commutative $R$-algebra over such a monoid $M$, denoted $R[M]$, as follows:
As an $R$-module, we define $R[M]=\bigoplus_{m\in M} R$, in other words $R[M]$ consists of formal finite sums of the form $\sum r_i X^{m_i}$ with $r_i \in R$ and $m_i \in M$. Here the symbol $X$ does not mean anything, I used it just in order for you to see the connection to ordinary polynomials. The multiplication is defined by extending the assignment $X^{m}\cdot X^n \mapsto X^{m+n}$, where $m+n$ is the operation in the monoid $M$, bilinearly to these formal sums (just like it is done in the case of usual polynomials). It is an easy exercise to see that for the monoid of natural numbers, this recovers the usual polynomial ring, i.e we find $R[\mathbb N]=R[X]$. If you however choose $M=\mathbb R$, you will get a polynomial ring with real exponents.