In [Geometric complexity theory][1] the [following][2] variety $\Delta[\text{det},m]$ is crucial.

 Let $X=(x_1,\ldots,x_r)$ be a tuple of $r=m^2$ variables, so that $X$ can be thought of as an $m\times m$ variable matrix,
identifying $x_i$'s with the entries of $X$ in any way.
By a homogeneous symbolic determinant of size $m$ over $X=(x_1,\ldots,x_r)$, we mean 
the determinant of a symbolic $m \times m$ matrix, whose each entry is a homogeneous linear function over $K$ of $x_1,\ldots,x_r$.
Let ${\cal X}$ be the 
vector space over $K$ of homogeneous polynomials of degree $m$ in the variables $x_1,\ldots,x_r$, and
$P({\cal X})$ the projective space associated with ${\cal X}$.
Let $\Sigma[\det,m] \subseteq P({\cal X})$ be the set of all points in $P({\cal X})$ 
that correspond to nonzero homogeneous polynomials in ${\cal X}$ that can be expressed as homogeneous 
symbolic determinants of size $m$ over $X$. 
Then $\Delta[\det,m] \subseteq P({\cal X})$ is  the Zariski-closure $\overline{\Sigma[\det,m]}$ of
$\Sigma[\det,m]$. Its dimension is $\le m^4$.

Consider $\Delta[\det,m]$ over $\overline{\mathbb{F}_q}$. Consider its zeta function $\zeta(s)$.

 By Dwork's theorem $\zeta(s) = \frac{p(s)}{q(s)}$ for some polynomials $p$ and $q$. 

My question: Is it true that $\deg(p)$, $\deg(q) \le 2^{poly(m)}$?




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If $\Delta[\det,m]$ were smooth, it would follow from Weil's conjectures. However, $\Delta[\det,m]$  [is not even normal][1] on $\mathbb{C}$.
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UPD: I am sorry I was wrong about my estimation of these degrees if $\Delta[\det,m]$ were smooth.


  [1]: http://www.unc.edu/math/Faculty/kumar/papers/kumar62.pdf
  [2]: http://math.stackexchange.com/questions/2171324/is-delta-det-m-smooth