In Jeffrey Lang, *A Jacobian identity in positive characteristic*, J. Commut. Algebra, Volume 7, Number 3 (2015), pp. 393--409, the following result is proven:

Theorem 1.Let $p$ be a prime. Let $\mathbf{k}$ be a commutative $\mathbb{F}_p$-algebra. Let $n$ be a nonnegative integer. Let $R$ be the polynomial ring $\mathbf{k}\left[x_1, x_2, \ldots, x_n\right]$. Let $f_1, f_2, \ldots, f_n$ be $n$ polynomials in $R$. Let $\nabla$ be the differential operator $\prod\limits_{i=1}^n \left(\dfrac{\partial}{\partial x_i}\right)^{p-1}$ on $R$. Let $M \in R^{n\times n}$ be the Jacobian matrix of $f_1, f_2, \ldots, f_n$; this is the $n\times n$-matrix over $R$ whose $\left(i,j\right)$-th entry is $\dfrac{\partial f_i}{\partial x_j}$. Then, \begin{align} & \sum\limits_{\left(i_1, i_2, \ldots, i_n\right) \in \left\{0,1,\ldots,p-1\right\}^n} f_1^{i_1} f_2^{i_2} \cdots f_n^{i_n} \nabla\left(f_1^{p-1-i_1} f_2^{p-1-i_2} \cdots f_n^{p-1-i_n} \right) \\ & = \left(-1\right)^n \left(\det M\right)^{p-1} . \end{align}

(I have taken the liberty to correct the typo in the paper where the sum ranged over $\left\{1,2,\ldots,p-1\right\}^n$ instead of $\left\{0,1,\ldots,p-1\right\}^n$. Note that later in the paper, in Proposition 1.5, there is a "$0 \leq j \leq n-1$" that should be a "$1 \leq j \leq n-1$"; this suggests blaming a miscommunication between author and editors about which $0$ to replace by a $1$.)

Theorem 1 generalizes Glynn's determinant formula (see Section 6 of Hendrik Lenstra, *The unit theorem for finite-dimensional algebras*, arXiv:1703.07273v1); indeed, it is easy to see that we can obtain the latter formula from Theorem 1 by setting $f_j = \sum\limits_{i=1}^n a_{ij} x_i$.

Question.Is there an elementary (e.g., combinatorial, inductive, or Hopf-algebraic) proof of Theorem 1?

The proof in Lang's paper relies on a different paper, which in turn relies on some algebraic geometry. I am not sure what is actually used in the proof, but the whole construct seems rather indirect.

One approach that looks promising is to consider the $\mathbf{k}$-linear map $\widetilde{\nabla} : R \otimes R \to R$ (all tensors are over $\mathbf{k}$) that sends each pure tensor $a \otimes b$ to $a \nabla\left(b\right)$. Then, the left hand side of Theorem 1 is \begin{equation} \widetilde{\nabla}\left( \prod_{i=1}^n \left(1 \otimes f_i - f_i \otimes 1\right)^{p-1} \right) \end{equation} (where the product is in $R \otimes R$). This is because of the classical fact that $\left(X-Y\right)^{p-1} = \sum\limits_{i=0}^{p-1} X^i Y^{p-1-i}$ in $\mathbb{F}_p\left[X,Y\right]$. But the question is whether $\widetilde{\nabla}$ has any good properties with respect to the Hopf algebra $R$.

Also, the operator $\nabla$ can be described rather explicitly on monomials: For any $n$ nonnegative integers $a_1, a_2, \ldots, a_n$, we have \begin{align} & \nabla \left(x_1^{a_1} x_2^{a_2} \cdots x_n^{a_n}\right) \\ & = \begin{cases} \left(-1\right)^n x_1^{a_1-\left(p-1\right)} x_2^{a_2-\left(p-1\right)} \cdots x_n^{a_n-\left(p-1\right)}, & \text{ if } a_i \equiv p-1 \mod p \text{ for all } i ; \\ 0, & \text{ otherwise }. \end{cases} \end{align}