Forms over finite fields and Chevalley's theorem

Question

Chevalley's theorem says that if $k$ is a finite field and $f(X_1,...,X_n)$ is a form (homogeneous polynomial) of degree $d < n$, then the equation $f(X_1,...,X_n) = 0$ has a non-trivial solution in $k^n$.

It is known that this result is optimal, in the sense that for each $n$ there exists a form $f(X_1,...,X_n)$ - coming from a norm - of degree $d = n$ which has only the trivial zero. See Brian Conrad's answer below.

I am interested in each form with this property, i.e. each form $f(X_1,...,X_n)$ of degree $d = n$ which has only the trivial zero. What are the known examples/classes of such forms? It would be very nice if we could describe/classify them all...

I am in particular interested in the case of quartic forms.

Answer 1

It is optimal for every $d > 0$. One way to make examples is using extension fields. For any $d > 0$, let $k'/k$ be an extension of degree $d$ and consider the norm map $N:k' \rightarrow k$. Choose a $k$-basis of $k'$, this is expressed as a homogeneous polynomial $f$ of degree $d$ in $d$ variables over $k$. More specifically, if $A$ is any $k$-algebra then the norm map $N_A:k' \otimes_k A \rightarrow A$ is given by evaluation of $f$ on $d$-tuples from $A$. This has no nonzero $k$-rational zeros, since nonzero elements of $k'$ have nonzero norm in $k$.

Answer 2

Perhaps it is obvious to most readers, but about a year ago I spent several days trying to determine for which pairs (d,n) there existed an anisotropic degree d form in n variables over a finite field $\mathbb{F}_q$. The question was motivated by Exercise 10.16 in Ireland and Rosen's classic number theory text: "Show by explicit calculation that every cubic form in two variables over $\mathbb{F}_2$ has a nontrivial zero."

As many students have discovered over the years, this is false: e.g. take

$f(x_1,x_2) = x_1^3 + x_1^2 x_2 + x_2^3$.

I knew about the existence and anisotropy of norm hypersurfaces for all $n = d$. But what about $n < d$? I confess that I spent some time proving this result in several special cases and even dragged a postdoc into it. Here is a copy of the sheepish email I sent out (in particular to Michael Rosen) later on:

If K is a field, and f(x_1,...,x_n) is an anisotropic form of degree d in n variables, then f(x_1,...,x_{n-1},0) is an anisotropic form of degree d in n-1 variables.

So let K be any field which admits field extensions of every positive degree d. Then for all d there is an anisotropic norm form N in d variables of degree d. For any n < d, setting (d-n) of the variables equal to 0 gives an anisotropic form of degree d in n variables. In particular, this proves "the converse of Chevalley-Warning".

So, not so fascinating after all, then.

I think it is still nontrivial to ask what happens if the hypersurface f is required to be geometrically irreducible. For instance, despite the fact that (q,3,3) is anisotropic, every geometrically irreducible cubic curve over a finite field has a rational point.

AS's question about classifying anisotropic hypersurfaces with $d = n$ is interesting. It may also be interesting to look at the case $d < n$. It is certainly not clear to me that all such anistropic hypersurfaces come from intersecting a norm hypersurface of larger dimension with a linear subspace.

I also want to add that the following generalization seemed less trivial to me (and I still don't know the answer): Chevalley-Warning is also true for sytems of polynomial equations $f_1(x_1,\ldots,x_n) = \ldots = f_r(x_1,\ldots,x_n)$ so long as the sum of the degrees of the $f_i$'s is strictly less than $n$. What kind of counterexamples can we construct here when $d = d_1 + \ldots + d_r \geq n$?