When, where and **how often** do you find polynomials of higher degrees than two in mathematical, pure/applied, research?

A formula for solving a polynomial of degree three, see this link; $ax^3+bx^2+cx+d=0$, is

$$\begin{align} x\quad&=\quad \sqrt[3]{ \left( \frac{-b^3}{27a^3} + \frac{bc}{6a^2} - \frac{d}{2a} \right) + \sqrt{ \left( \frac{-b^3}{27a^3} + \frac{bc}{6a^2} - \frac{d}{2a} \right) ^2 + \left( \frac{c}{3a} - \frac{b^2}{9a^2} \right) ^3 } }\\ &+\quad \sqrt[3]{ \left( \frac{-b^3}{27a^3} + \frac{bc}{6a^2} - \frac{d}{2a} \right) - \sqrt{ \left( \frac{-b^3}{27a^3} + \frac{bc}{6a^2} - \frac{d}{2a} \right) ^2 + \left( \frac{c}{3a} - \frac{b^2}{9a^2} \right) ^3 } } \;-\;\frac{b}{3a} \end{align}$$

Unlike quadratic, cubic, and quartic polynomials, the general quintic cannot be solved algebraically in terms of a finite number of additions, subtractions, multiplications, divisions, and root extractions, as rigorously demonstrated by Abel (Abel's impossibility theorem) and Galois. However, certain classes of quintic equations can be solved [...] Source: http://mathworld.wolfram.com/QuinticEquation.html

At levels of $5^{\text{th}}$ degree polynomials, things are starting to look really serious in my eyes. My question is:

If it is possible to not answer subjectively: When, where and how often do you find polynomials of higher degrees than two in mathematical, pure/applied, research?

Solution 1:

The characteristic polynomial of an $n \times n$ matrix has degree $n$. We often care about matrices larger than $2 \times 2$.

Solution 2:

It's actually (somewhat) rare in research that you are lucky enough to have a polynomial that is only degree one or two. Some examples I haven't seen mentioned already: cyclotomic polynomials, irreducible polynomials used to construct field extensions, all sorts of polynomials used to construct algebraic curves used in cryptography.

While we don't always have nice formulas to explicitly find the roots, we have other ways to work with them. For example, we can choose nice polynomials that have a special form, or construct a polynomial so we already know the roots. We can use computational techniques to approximate the roots. We can use them in applications where we don't care what the roots are.

Solution 3:

Cubic polynomials are ubiquitous in computer-aided design and computer graphics.

They also are the basis for computer fonts.

Finite element analysis is based on polynomial functions. Isogeometric analysis uses NURBS.

Solution 4:

From the applied math side of things, I come across higher order polynomials all the time. Here are some examples of when:

Eigenvalue problems for large matrices

Linear constant coefficient higher-order ODEs (think characteristic equation, or Laplace transform to transform ODE to solving a polynomial)

Numerical analysis - interpolation, cubic splines, higher accuracy integration schemes

Many times we expand analytic functions by their Taylor series, which as you know are polynomials of whatever degree we choose to stop expanding at.

Polynomials are ubiquitous throughout probably every field of higher mathematics :)