Definition of polynomial

Question

In the wikipedia, they define polynomial as follows:

In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables.

Then if is there operations of differential, integral, etc. in the expression (for example)

$1+\frac{dy}{dx}$

then aren't expressions which have differential or integration like above can't be considered polynomial?

Answer 1

Perhaps the concept you are looking for is of differential polynomials.

A differential polynomial over a differential field $K$ is a formalization of the concept of differential equation such that the known functions appearing in the equation belong to $K$ and the indeterminates are symbols for the unknown functions.

This extends the idea of a polynomial to include differentiation operations (actually termed "derivations") applied to variables.

A more precise definition of polynomial from Wikipedia is in polynomial ring

In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, often a field.

However, while this definition is useful in abstract algebra, it is not what is commonly used in practice. A key idea is that of a polynomial expression relative to some given set of other expressions. That is, for example, $\cos(x)^2 + \sin(x)^2 - 1$ is a polynomial expression relative to $\{\cos(x),\sin(x)\}.$ Thus, another example is that $1 + \frac{dy}{dx}$ is a polynomial expression relative to $\{\frac{dy}{dx}\}.$

Answer 2

The key is that there is a difference between a polynomial and a polynomial function.

Of particular importance is the notion of an indeterminate.

$1+z$ is a polynomial, with $z$ an indeterminate. $1+\frac{dy}{dx}$ is an evaluation of the related polynomial function at $\frac{dy}{dx}.$

The distinction between polynomial and polynomial function is somewhat (but not entirely) irrelevant when talking about polynomials with real or complex coefficients, because, given two such polynomials, they are equal if and only if the functions are always equal.

But with coefficients over finite rings, or rings with zero divisors, it is possible that two different polynomials can be associated with the same polynomial function on that ring.

The prototypical case is the ring $R=\mathbb Z/p\mathbb Z,$ where $p$ is prime. Then $f(x)=x^{p}$ and $g(x)=x$ are two different polynomials but they are equal as polynomial functions on $R.$

So mathematicians define a concept of "indeterminate." Given a commutative ring $R,$ we can define rings $R[x]$ or $R[x,y,z]$ or $R[x_i]_{i=1}^{\infty}.$ These rings are the rings of polynomials, with, respectively, indeterminate $x$, or $x,y$ and $z,$ or $x_1,x_2,\dots.$

In these rings, we often write polynomials like $f(x,y,z)=x^2+2y^2+3z^2,$ looking like function definitions, but the variables are not meant to represent specific values. They are abstractions.

Now, given a polynomial $f\in R[x_1,\dots,x_n],$ we also have a polynomial function $\overline f:R^n\to R,$ which assigns to each $(r_i)_{i=1}^n\in R^n$ a value equal to what you would think of as the evaluation of that polynomial on $(r_1,\dots,r_n).$

But even more generally, given a homomorphism $R\to S,$ where $S$ is another commutative ring, we also get a polynomial function $\overline f:S^n\to S.$

For a simple example, if $R$ is the ring $\mathbb Z$ of integers, and $S$ is the ring $\mathbb C,$ of complex numbers, then we can evaluate $f(x,y)=2x+4xy+y^2\in\mathbb Z[x,y]$ as a polynomial function of complex numbers.

So a single polynomial actually can define a lot of functions.

Rings of polynomials are interesting, because they are the "simplest" commutative ring which contains which contain $R$ and the indeterminates. This can be made rigorous by showing a certain "universal property," but you can think of it as the smallest commutative ring of expressions formed by elements of $R$ and the indeterminates, using $+$ and $\times$ as your only operation.

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(Note that, if we have infinitely many indeterminates, each polynomial will still only be a finite expression, and thus cannot use all the indeterminates.)