An Analog of Determinant Related to Parshin–Kato Theory and Integer Polytopes

Parshin–Kato theory involves a multilinear function of $n+1$ vectors in the $n$-dimensional vector space over the field $\mathbb{Z}/2\mathbb{Z}$. The same function arises in the computation of the product in the group $(\mathbb{C}^*)^n$ of all roots of several polynomial equations with sufficiently generic Newton polytopes. We discuss this remarkable function and related geometry of integer polytopes.