What are the conditions for a polynomial?

Question

I know this is an extremely basic and simple question, but I have googled it and looked at some of the responses but I still don't get the exact conditions for an expression(s) to be considered a polynomial. From what I understood:

1. Polynomials must have a variable raised to the power of whole numbers, it cannot have rational or a negative number as a power.

2. There must be no variable in the denominator, to which I'm quite confused, because does that make the expression, for example, x^2 / x to be a non-polynomial expression?

3. There must be no non-basic operation in the expression, such as log, any of the trig functions, ln, etc. But I'm confused about this, is having a log or a trig function as a coefficient of a variable allowed? Or is it simply not allowed in the expression at all.

4. There must be no variable in the exponent, ie 2^x or 1.6^y

That's most of the conditions I got, not sure if there are any more important ones that are missing. Are these conditions true? Is there a much simpler method of figuring out a certain expression is a polynomial? Is it extremely important to classify polynomials and non-polynomials, or is it just something nice to know?

Answer 1

It becomes a bit clearer when we talk about an expression being polynomial in a given set of variables. Once we have established that basis, then the expression is polynomial if those variables are only ever multiplied together with each other or a constant value, or added a finite number of times.

So then a polynomial in $x_1, x_2, \ldots, x_m$ is an expression that can be written as

$$\sum_{j = 1}^N a_j x_1^{n_{1j}} x_2^{n_{2j}} \ldots x_m^{n_{mj}}$$

where the $a_j$ values are being considered as constant for the purpose of this expression, and the $n_{ij}$ are non-negative integers.

This means that $x y$ is a polynomial in $x$ and $y$, as is $3 x^2 + 4 y^3$, but $x^y$ is not. If you know that $y$ is a natural number then you could say that $x^y$ is a polynomial in $x$ of degree $y$. $\sqrt{2} xy$ is also polynomial in $x$ and $y$, as is $\sqrt{L} xy$, but it is not polynomial in $L$.

This also means that our variables could also be more complex expressions. For example, I would say that $5 + 3 \log x + 7 (\log x)^2$ is polynomial in $\log x$, even though it isn't polynomial in $x$.

For that reason, I would not say that $\frac{x^2}{x}$ is a polynomial expression, even though for $x \neq 0$ it simplifies to a polynomial. In some contexts the difference might not matter, and sometimes it might be enough to talk about whether an expression is equivalent to a polynomial even if it is not one itself.

As for why we bother, polynomials have a lot of nice properties that don't always extend to other functions. They are continuous, smooth and entire, which are all incredibly useful properties in real and complex analysis. They often factorise in particular ways that we can make use of. The Fundamental Theorem of Algebra says that a single-variable polynomial with complex coefficients will have exactly as many roots as its degree (including multiplicity), which provides an important categorisation of the complex numbers as an algebraically closed field. The way that polynomials are constructed as sums of monomials means that you can treat the set of polynomials with maximum degree $N$ as a vector field with those monomial terms as the basis, which then leads to things like polynomial interpolation.