An analog of determinant related to Parshin-Kato theory and integer polytopes (Q2498294)

The main object of the paper under review is a unique function \(D(k_1,\dots ,k_{n+1})\) defined on sets of \((n+1)\) vectors in the \(n\)-dimensional vector space over \({\mathbb Z}/2\mathbb Z\) with values in \({\mathbb Z}/2\mathbb Z\) which satisfies the following properties: (a) \(D(k_1,\dots ,k_{n+1})=0\) if the rank of the system of vectors \(k_1,\dots ,k_{n+1}\) is less than \(n\); (b) \(D(k_1,\dots ,k_{n+1})=\lambda_1+\dots +\lambda_{n+1}+1\) if the vectors \(k_1,\dots ,k_{n+1}\) satisfy the single relation \(\lambda_1k_1+\dots +\lambda_{n+1}k_{n+1}=0.\) This function, which can be viewed as an analog of determinant, is multilinear and \(\text{GL}_n({\mathbb Z}/2\mathbb Z)\)-invariant. After presenting several explicit formulas for the function \(D\) in terms of classical determinants, the author discusses geometry related to this function. Namely, the main result of the paper, theorem 4, relates \(D\) to the antiderivative of the volume form on the \((n+1)\)-dimensional torus. As a by-product, the author obtains some new results in geometry of integer polytopes (theorems 3 and 3'). He also indicates that the function \(D\) arises in several different contexts (systems of polynomial equations with sufficiently general Newton polytopes; multidimensional resultants; the Parshin--Kato class field theory).