Question

Let $\mathbb F_q$ be a finite field with $q$ elements where $q$ is a power of a prime $p$. Consider the polynomial ring

$$\mathbb F[x_{ij}],$$ for $i,j=1,\ldots ,n$. Let $f$ and $g$ be polynomils in $\mathbb F[x_{ij}]$ defined by $$f=\sum_{\sigma\in S_n}sgn(\sigma)x_{1\sigma(1)}\ldots x_{n\sigma(n)}$$ and $$g=\sum_{\sigma\in S_n}x_{1\sigma(1)}\ldots x_{n\sigma(n)},$$ where $S_n$ denotes the set of all permutations of the set {1, 2, . . . , n} and $sgn$ is the signum of the permutation $\sigma$.

Let $X$ and $Y$ be the projective homogeneous varieties defined by $f$ and $g$, respectively.

how can I prove (I feel that it's true) that the varieties $X$ and $Y$ have not the same number of points?

Answer 1

In a recent work http://arxiv.org/abs/1003.1984 the authors considered the sets $$P_n(\mathbb{F}) = \lbrace A \in \mathbb{F}^{n \times n} \mid g(A) = 0 \rbrace$$ $$D_n(\mathbb{F}) = \lbrace A \in \mathbb{F}^{n \times n} \mid f(A) = 0 \rbrace$$

They showed $|P_3(\mathbb{F})| = |D_3(\mathbb{F})| - q^2(q-1)^5 < |D_3(\mathbb{F})|$.

In case $n> 3$ they showed that there is a constant $k=k(n)$ such that for all $\mathbb F$ with $q > k$, $|P_n(\mathbb{F})| < |D_n(\mathbb{F})|$ holds.

Consequently, the same is true for $|V(g)|= |P_n(\mathbb{F})|/(q-1)$ and $|V(f)|= |D_n(\mathbb{F})|/(q-1)$.