Why are polynomials interesting?

Question

I'm learning polynomials (for competitions) the first time, with having a little number theory and combinatorics experience.

The difficulty is that I can't understand

1. why polynomials are so important, i.e why a very special class of functions $f(x) =\sum_{i} a_ix^i$ is very important in mathematics

2. why I should care about factoring a polynomial into shorter polynomials with integer coefficients, or expressing it as $\sum_{i} a_ix^i = \Pi_{i} (x + b_i)$ (equivalently, finding the zeroes of it)?

3. What are the application of polynomials in other apparently different fields of mathematics?

(Also as a bonus question, How to think about/imagine them from a non-symbolic, intuitive and/or visual perspective(s) ?)

Answer 1

If you accept that arithmetic (addition and multiplication) is interesting (and fundamental to mathematics) then polynomials are a simple way to write down rules for processes of addition and multiplication.

Then one can observe that there are simple rules for manipulating these processes, which amounts to doing algebra with polynomials. Moreover, there is lots of structure inside the processes themselves (factorization, derivatives, Galois theory, algebraic geometry, representation theory, invariant theory, etc.).

Let's return to the idea of polynomials as description of a process for doing arithmetic with numbers.

For example, $x^2 + 1$ expresses the process of squaring a number, and adding one. For real $x$ , this expression never less than 1, so has no roots. However, the catch was in the description of a polynomial as "a process for doing arithmetic with "numbers.""

What numbers are we considering? We could have plugged in only whole numbers instead of the reals and features of this process would be different (e.g. the output would be whole numbers, and have "jumps"). (So perhaps there are some things that are intrinsic to the process, and some things that depend on what numbers you are "plugging in" to it. This point of view turns out to be very fruitful ... for more on this, see the article by Jordan Ellenberg in the Princeton Guide to Pure Mathematics: http://press.princeton.edu/chapters/gowers/gowers_IV_5.pdf )

As you (may) know, we can construct the complex numbers, where the "process" $x^2 + 1$ can result in $0$, by plugging in $i$ or $-i$.

Moreover, one can build the complex numbers using polynomials with real coefficients. The key observation is that, like numbers, you can add and multiply polynomials, and you can also multiply polynomials by a real number. So given any polynomial $p(x)$, you can plug a polynomial into it to obtain a new polynomial.

If I declare the polynomial $t^2 + 1$ to be $0$ (just go with it for now), then when I plug $t$ into $x^2 + 1$, I get $t^2 + 1$, which I have said is zero. So, if I study all polynomials of the form $\Sigma a_i t$,with $a_i$ real, I appear to have a solution to the equation $x^2 + 1$. (This means that I have a number,"$t$", that I can input into the process $x^2 + 1$ a.k.a "square and add one", and get out the thing which I call $0$.)

So $t$ behaves like $i$, and indeed one can prove that this system of polynomials has all the properties that we want the complex numbers to have. From an operational point of view (the properties we want some object to have), we have constructed the complex numbers out of polynomials and real numbers.

For (much) more on this, you can try to read Ian Stewart's Galois Theory. (It will be hard, but you shouldn't be afraid. If I recall correctly, it doesn't assume much by way of prerequisites, but you may need to consult some other texts on abstract algebra along the way.)

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Lastly -- for a visual point of view on polynomials, I find it helpful (in some situations, certainly not for computation), to identify polynomials with their roots. I can then imagine the roots floating around on the real line (or complex plane). (Note that this point of view does not distinguish $x^2$ and $2x^2$, but does distinguish $x^2$ from $x$, if you imagine that there are two points at zero when visualizing $x^2$.) This leads to the notion of configuration space - the "space" of points in some other "space." (And also has connections to Galois theory, via "covering" spaces, when you try to imagine what happens to the roots as we move the coefficients of the polynomials around... for example, try this for $z^2 - a$, as you move $a$ clockwise on the unit circle in the complex plane.)

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Answer 2

Some thoughts.

A)

Polynomials are fundamental to numbers because if $n \in \mathbb{N}$ then (without necessarily knowing it) we are actually used to thinking of the decimal digits of $n$ as the coefficients in a polynomial and then the number $n$ is the result of evaluating that polynomial at `$10$'. i.e.

$$ n = \sum_{j=0}^{+\infty} d_j 10^j $$ with all but finitely many of the $d_j$ being non-zero. Thus $$ n = (d_0 + d_1 x + \dots + d_N x^N) \rvert_{x = 10} $$ where $N+1$ is the number of digits of $n$. You can of course do this for any base.

B)

Of course you know that if you can factor a polynomial $P = \prod (x - r_j)$, then $P(x) = 0$ precisely when $x = r_j$. And so factorizing polynomials is solving polynomial equations. And solving equations would appear to be one of the most basic things that math is about, right?

Are all real numbers the root of some polynomial equation with integer coefficients? (NO: https://en.wikipedia.org/wiki/Transcendental_number)

Also, this is nice: $x^{105} -1$ has a factor that has a coefficient of $2$. For a while all factors of $x^n - 1$ were thought to only contain coefficients that were absolute value 1. (asked about here Maximum absolute value of polynomial coefficients)

Even more interesting is solving equations over specific fields: Take a polynomial of degree $n$. It is a significant theorem that there are exactly $n$ complex roots when counted with multiplicity, right? (It's called the Fundamental Theorem of Algebra and there are many different proofs: See this highly upvoted MO question

https://mathoverflow.net/questions/10535/ways-to-prove-the-fundamental-theorem-of-algebra

) That's a theorem with real mathematical content!

It's not even trivial that there are at most $n$ roots! (sure if $(x-a)$ is a factor then $a$ is a root, but if $a$ is a root, then is it really obvious from first principles that $P/(x-a)$ is a polynomial? I would say no).

Understanding multivariate polynomials over the rational numbers is basically a huge theme of number theory, This includes ridiculously difficult problems such as Fermat's Last Theorem ($x^n + y^n - z^n = 0$) and more generally understanding the number of rational points on elliptic curves.

C)

All of this is not to mention much about the geometry of polynomials. One angle is the geometry of the roots or the values of the polynomial, e.g.

Look up Madelbrot sets for the incredible fractal geometries obtained by iterating polynomials as simple as $$ z \mapsto z^2 + c $$ in the complex plane.

Or - another hugely upvoted MO question - Why is this the geometric picture you see when you plot roots of random polynomials??

https://mathoverflow.net/questions/182412/why-do-roots-of-polynomials-tend-to-have-absolute-value-close-to-1

A more advanced angle again is multivariate polynomials. We get into algebraic geometry pretty quickly:

Just consider this fundamental geometric object $$\{x^2 + y^2 - 1 = 0\}$$ the circle! It is the roots of a certain polynomial! What other shapes can you get just from two variables and degree two??? e.g. what is $$\{xy + y^2 - x = 0\}$$?? Or how about in higher dimensions? e.g. $$\{x^2 + y^2 - z + 3yz = 0\}$$?What is the topology of that surface? How does the topology of the surface relate to the degree of the polynomial?

I don't like to exaggerate too much but THE LIST IS ENDLESS