Geodesic
An Analog of Determinant Related to Parshin--Kato Theory and Integer Polytopes
Funkcionalʹnyj analiz i ego priloženiâ, Tome 40 (2006) no. 2, pp. 55-64.

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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. 
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A. G. Khovanskii. An Analog of Determinant Related to Parshin--Kato Theory and Integer Polytopes. Funkcionalʹnyj analiz i ego priloženiâ, Tome 40 (2006) no. 2, pp. 55-64. http://geodesic.mathdoc.fr/item/FAA_2006_40_2_a5/

[1] Khovanskii A., “Newton polyhedrons, a new formula for mixed volume, product of roots of a system of equations”, The Arnoldfest, Proceedings of a conference in honour of V. I. Arnold for his 60th birthday (Toronto, Canada, June 15–21, 1997), Fields Inst. Commun., 24, Amer. Math. Soc., Providence, RI, 1999, 325–364 | MR | Zbl

[2] Parshin A. N., “Lokalnaya teoriya polei klassov”, Trudy MIAN, 165, 1984, 143–170 | MR | Zbl

[3] Parshin A. N., “Kogomologii Galua i gruppa Brauera lokalnykh polei”, Trudy MIAN, 183, 1990, 159–169 | MR

[4] Khovanskii A. G., Logarifmicheskii funktsional i mnogomernye simvoly, Gotovitsya k pechati