In Geometric complexity theory the following 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 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.