# Convex bodies and algebraic equations on affine varieties - Mathematics > Algebraic Geometry

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Abstract: Given an affine variety X and a finite dimensional vector space of regularfunctions L on X, we associate a convex body to X, L such that its volume isresponsible for the number of solutions of a generic system of functions fromL. This is a far reaching generalization of usual theory of Newton polytopeswhich is concerned with toric varieties. As applications we give new, simpleand transparent proofs of some well-known theorems in both algebraic geometrye.g. Hodge Index Theorem and convex geometry e.g. Alexandrov-Fenchelinequality. Our main tools are classical Hilbert theory on degree ofsubvarieties of a projective space in algebraic geometry and Brunn-Minkowskiinequality in convex geometric.

Author: ** Kiumars Kaveh, Askold G. Khovanskii**

Source: https://arxiv.org/