Quadratic forms play a huge role in math. This leads one to wonder: Is there a theory of cubic forms, quartic forms, quintic forms and so on? I have failed to discover any. Is there any such theory? If not, is it because:


* It is not as interesting as quadratic forms?
* It is so hard that no-one has yet written about such a theory?
* It is already deeply infiltrated in math and only some smart people know about it?



There is a nice theory of quadratic forms. How about cubic forms, quartic forms, quintic forms, ...?
I once asked André Weil the same question.


When I was college, taking a course that discussed quadratic forms, Weil gave a guest lecture to the students about that topic. After the talk, I raised my hand and asked him why there was such a big deal in math about quadratic forms while it seemed there was nothing comparable for higher-degree forms. Weil gave an answer, but to my regret I could not understand it (difficulty hearing him) and I did not ask him later to repeat what he had said. Now many years later, I can offer an answer that I think my former student self would have found satisfactory.


Before I begin, let me point out that we all know one important higher-degree form: the determinant form of degree $n$. So it is reasonable to ask what kind of general theory there could be for higher degree forms.


First let's see that the bijection between quadratic forms and symmetric bilinear forms generalizes to higher degree.
Recall for a field $F$ not of characteristic $2$, there is a bijection between quadratic forms $Q : F^n \to F$ and symmetric bilinear forms $B : F^n \times F^n \to F$ by $Q(\mathbf x) = B(\mathbf x,\mathbf x)$ and
$$
B(\mathbf x,\mathbf y) = \frac{1}{2}(Q(\mathbf x + \mathbf y) - Q(\mathbf x) - Q(\mathbf y)).
$$ Replacing $2$ by a degree $d \geq 1$, for a field $F$ where $d! \not= 0$ (meaning $F$ has characteristic $0$ or characteristic $p$ for $p > d$) there is a bijection between forms $f : F^n \to F$ of degree $d$ and symmetric $d$-multilinear maps $\Phi : \underbrace{F^n \times \cdots \times F^n}\_{d \ {\sf copies}} \to F$ where $f(\mathbf x) = \Phi(\mathbf x,\ldots,\mathbf x)$ and
$$
\Phi(\mathbf x\_1,\ldots,\mathbf x\_d) = 
\frac{1}{d!}\sum\_{\substack{J \subset \{1,\ldots, d\} \\ J \not= \emptyset}} (-1)^{d - |J|}f\left(\sum\_{j \in J} \mathbf x\_j\right), 
$$
(You could include $J = \emptyset$ in the sum by the usual convention that an empty sum is $\mathbf 0$, since $f(\mathbf 0) = 0$.) For example, when $f$ is a cubic form ($d = 3$, $n$ arbitrary), the associated symmetric trilinear form is
$$
\Phi(\mathbf x,\mathbf y,\mathbf z) = \frac{1}{6}(f(\mathbf x + \mathbf y + \mathbf z) - f(\mathbf x + \mathbf y) - f(\mathbf x + \mathbf z) - f(\mathbf y + \mathbf z) + f(\mathbf x) + f(\mathbf y) + f(\mathbf z)).
$$
For example, if $f : F^3 \to F$ by $f(x\_1,x\_2,x\_3) = x\_1^3+x\_2^3+x\_3^3$ then $\Phi(\mathbf x,\mathbf y,\mathbf z) = x\_1y\_1z\_1 + x\_2y\_2z\_2+x\_3y\_3z\_3$.
The general formula for $\Phi$ in terms of $f$
shows why we want $d! \not= 0$ in $F$.
Over fields of characteristic $0$, I think this bijection is due to Weyl.


Using this bijection, we call a form $f$ of degree $d$ *nondegenerate* if, for the corresponding symmetric multilinear form $\Phi$, we have
$\Phi(\mathbf x,\mathbf y, \ldots, \mathbf y) = 0$ for all $\mathbf y$ in $F^n$ only when $\mathbf x = \mathbf 0$. (Equivalently,
we have $\Phi(\mathbf x,\mathbf x\_2, \ldots, \mathbf x\_d) = 0$ for all $\mathbf x\_2, \ldots, \mathbf x\_d$ in $F^n$ only when $\mathbf x = \mathbf 0$.) When $d = 2$ (the case of quadratic forms), this is the usual notion of a nondegenerate quadratic form (or nondegenerate symmetric bilinear form).


That the bijection between quadratic forms and symmetric bilinear forms can be extended to higher degrees suggests there might be general theory in higher degree that's just like the quadratic case, but it turns out there really are significant differences between quadratic forms and forms of higher degree. Here are two of them.


1. Diagonalizability.


Outside characteristic 2, a quadratic form can be diagonalized after a linear change of variables, but for $n \geq 3$, a form of degree $n$ might not be diagonalizable after any linear change of variables.
While any nondegenerate *binary* cubic form over $\mathbf C$ can be diagonalized (see the start of the proof of Lemma 1.7 [here](https://arxiv.org/pdf/2103.16691.pdf); in the binary case, nondegeneracy of a cubic form is equivalent to the dehomogenization being a cubic polynomial with nonzero discriminant), nondegenerate cubic forms over $\mathbf C$ in more than two variables need not be diagonalizable. For example,
the three-variable cubic form $x^3 - y^2z - xz^2$ is nondegenerate and can't be diagonalized over $\mathbf C$ for a reason related to elliptic curves: see my comments on the MO pages [here](https://mathoverflow.net/questions/238901/cubic-forms-and-finiteness-of-k-k3) and [here](https://mathoverflow.net/questions/77702/transformation-of-a-cubic-form).
For each $d \geq 3$ and $n \geq 2$ *except* for $d=3$ and $n = 2$, there are nondegenerate forms of degree $d$ in $n$ variables over $\mathbf C$ that are smooth away from $(0,0,\ldots,0)$ and are not diagonalizable. Note the diagonal form $x\_1^d + \cdots + x\_n^d$ is smooth away from the origin.


2. Group theory.


For a form $f(x\_1,\ldots,x\_n)$ over a field $F$, its *orthogonal group* is the linear changes of variables on $F^n$ that preserve it:
$$
O(f) = \{A \in {\rm GL}\_n(F) : f(A\mathbf v) = f(\mathbf v) \ {\rm for \ all } \ \mathbf v \in F^n\}.
$$
Nondegenerate quadratic forms have a rich orthogonal group (many reflections) and some higher-degree forms have a large orthogonal group: if $f$ is the determinant form of degree $n$ then its orthogonal group is ${\rm SL}\_n(F)$.
But for a form $f$ of degree $d \geq 3$ over an algebraically closed field of characteristic $0$, $O(f)$ is sometimes a *finite* group. This happens if the corresponding symmetric $d$-multilinear form $\Phi$ satisfies $\Phi(\mathbf x, \ldots, \mathbf x,\mathbf y) = 0$ for all $\mathbf y$ in $F^n$ only when $\mathbf x = 0$. When $d \geq 3$ this condition is different from nondegeneracy as defined above. Let's say such $f$ and $\Phi$ are *nonsingular*. That nonsingular forms of degree $d$ have a finite orthogonal group over $\mathbf C$ is due to Jordan.
It also holds over algebraically closed fields of characteristic $p$ when $p > d$ (so $d! \not= 0$ in the field).
As an example, the orthogonal group of $x\_1^d + \cdots + x\_n^d$ over $\mathbf C$ when $d \geq 3$ has order $d^n n!$: it contains only the compositions of $n!$ coordinate permutations and scaling of each of the $n$ coordinates by $d$th roots of unity. Taking $n = 2$, this reveals a basic difference between the concrete binary forms $x^2 + y^2$ and $x^d + y^d$ for $d \geq 3$ that you can tell anyone who asks you in the future how higher degree forms are different from quadratic forms.


In retrospect, the label used for the second topic ("Group theory") really applies to both topics. For a field $F$,
the group ${\rm GL}\_n(F)$ acts on the forms of degree $d$ in $n$ variables with coefficients in $F$, and the first topic is about the orbit of $x\_1^d + \cdots + x\_n^d$ under this action while the second topic is about the stabilizer of $f(x\_1,\ldots,x\_n)$ under this action.


Concerning papers and books, I'll just mention one of each. There is Harrison's paper "A Grothendieck ring of higher degree forms" in J. Algebra 35 (1978), 123-138 [here](https://core.ac.uk/download/pdf/82089961.pdf) and Manin’s book *Cubic forms: algebra, geometry, arithmetic*. Manin mentioned a recurring nightmare he had about this book, soon after he finished it, in an interview with Eisenbud [here](https://www.youtube.com/watch?v=2wGRUBlsizE?t=177).


