What a function does to a space of functions

Write down a polynomial of degree at most five and you have written down six numbers. The polynomial $0.1x^5 - 0.55x^3 + 0.9x$ is the list $(0, 0.9, 0, -0.55, 0, 0.1)$, read off the powers $1, x, x^2, \ldots, x^5$, and every polynomial of degree at most five is exactly one such list. Adding two of them adds the lists. Scaling one scales the list. So the polynomials of degree at most $n$ are a vector space of dimension $n+1$, no more mysterious than $\mathbb{R}^6$, and the monomials $x^k$ are its coordinate axes.

That reframing is worth more than it looks. A function on the real line is an infinite amount of information, one value per point, and there is no obvious sense in which one function is close to another or built out of others. A point in $\mathbb{R}^6$, on the other hand, is something you already know how to push around. And the things we do to polynomials all day — shift them, stretch them, differentiate them, multiply them by $x$, feed one into another — are maps from that space to itself, or to some other space just like it. They have matrices. You can look at them.

interactive · five operators on the polynomials of degree ≤ 5open ↗

Pick an operator from the dropdown and watch the matrix. Shifting by $h$, sending $p(x)$ to $p(x+h)$, fills in an upper triangle whose diagonal is all ones. Scaling, $p(x) \mapsto p(ax)$, is diagonal, with $a^k$ down the diagonal. The derivative is a single stripe sitting just above the diagonal. Multiplying by $x$ is a stripe just below it, and its matrix is not even square — it takes a six-dimensional space into a seven-dimensional one, because it has to. Substituting $x^2$ pushes every coefficient twice as far out.

Every one of those is a linear map of $p$. That is what having a matrix means, and it is a slightly surprising fact, because none of these operations is linear in $x$. Differentiation obeys $(p+q)' = p' + q'$. Shifting obeys $(p+q)(x+h) = p(x+h) + q(x+h)$. Even substituting $x^2$, which does something violent to the graph, does it additively: $(p+q)(x^2) = p(x^2) + q(x^2)$. Linearity in the function and linearity in the variable are different questions, and confusing them is the single most common way to get lost here.

So if all of them are linear, what actually distinguishes them? Look at the degree readout, not the matrix. The shift and the scale leave the degree exactly where it was. The derivative eats one. Multiplication by $x$ adds one. Substituting $x^2$ doubles it. The degree is the thing that moves, and it moves in patterns that are much more rigid than the matrices suggest.

Degree is a strange quantity. It is not linear — $\deg(p+q)$ is usually $\max(\deg p, \deg q)$ but drops without warning when leading terms cancel. It is not continuous: slide the leading coefficient of a quintic to zero and the degree falls off a cliff from 5 to 4 at the last instant. What it is, is a labelling of a nested chain of subspaces,

$\mathcal{P}_0 \subset \mathcal{P}_1 \subset \mathcal{P}_2 \subset \cdots \subset \mathcal{P}_n,$

constants inside linear inside quadratic and so on, each one a hyperplane in the next. Saying $\deg p = 3$ says that $p$ lives in $\mathcal{P}_3$ and not in $\mathcal{P}_2$. That chain is called a flag, and an operator preserves degree precisely when its matrix is triangular in the monomial basis with nothing vanishing on the diagonal. Triangular is not a computational accident. It is what “respects the flag” looks like when you write it down.

There is a second structure on this space that the vector-space picture completely ignores, and it is the one that makes polynomials interesting rather than merely six-dimensional. You can multiply two of them. You can also compose two of them. Neither operation is available in $\mathbb{R}^6$, neither is linear, and the two of them treat degree in different ways — one adds, the other multiplies. A space with two incompatible multiplications on it and a grading that each one respects differently is a much richer object than a list of numbers, and almost everything that follows is a consequence of those two laws colliding.

That is the series. First, degree is a coordinate: what the matrix of $p(x) \mapsto p(ax+b)$ looks like, why it is triangular, and why an invertible affine substitution cannot possibly change a polynomial’s degree. Then products add, composition multiplies, which is the pair of laws just mentioned, and which turns out to prove — in one line, with no calculation — that the only polynomials with polynomial inverses are the affine ones.

Then the two operators that break the symmetry. The derivative is nilpotent, dying after $n+1$ applications, and its exponential $e^{hD}$ is the shift, which is Taylor’s theorem wearing a matrix. And the projective fix, where we allow $(ax+b)/(cx+d)$, watch it tear a pole in the graph, and repair everything by admitting that the missing structure was a point at infinity all along.

Then log paper and the shape of a law. Change coordinates on the domain instead of acting on the function, and multiplication becomes addition, powers become slopes, and the integer degree we spent five articles protecting relaxes into a real number. That number is a fractal dimension, and it is the reason the rest of this site exists.

After that, six more articles in which none of the algebra changes and all of the vocabulary does. Give the polynomial two more variables and ask which ones the Laplacian kills. The answer is a space of dimension $2\ell+1$: $\ell$ is a degree, that $2\ell+1$ is the number of orbitals in a subshell, and the shapes printed in every chemistry textbook are the sign regions of homogeneous polynomials in $x$, $y$, $z$. The same polynomials, sorted by degree, are the terms of the multipole expansion, each decaying one power faster than the last. And electrostatics, the hydrogen atom, and the quantum oscillator — three different derivations in three different chapters — turn out to be one angular polynomial times three different radial transcendentals, with the potential getting a vote on the second factor only.

Then what an external field does to that grading, which is most of atomic spectroscopy read as polynomial algebra. An electric field is multiplication by $z$, a harmonic of degree one, and the selection rule $\Delta\ell = \pm1$ is nothing but degrees adding. A magnetic field is a diagonal operator whose eigenvalue is the label you had already written on the state. And a crystal field is the lowest-degree polynomial a cube is permitted to have, which turns out to be degree four, which is why the five $d$ orbitals split into two and three, and why a ruby is red.