{
  "id": "0804.4095",
  "version": "v1",
  "published": "2008-04-25T11:57:50.000Z",
  "updated": "2008-04-25T11:57:50.000Z",
  "title": "Convex bodies and algebraic equations on affine varieties",
  "authors": [
    "Kiumars Kaveh",
    "Askold G. Khovanskii"
  ],
  "comment": "Preliminary version, may contain several typos, 44 pages",
  "categories": [
    "math.AG"
  ],
  "abstract": "Given an affine variety X and a finite dimensional vector space of regular functions L on X, we associate a convex body to (X, L) such that its volume is responsible for the number of solutions of a generic system of functions from L. This is a far reaching generalization of usual theory of Newton polytopes (which is concerned with toric varieties). As applications we give new, simple and transparent proofs of some well-known theorems in both algebraic geometry (e.g. Hodge Index Theorem) and convex geometry (e.g. Alexandrov-Fenchel inequality). Our main tools are classical Hilbert theory on degree of subvarieties of a projective space (in algebraic geometry) and Brunn-Minkowski inequality (in convex geometric).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-04-25T11:57:50.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "convex body",
      "affine variety",
      "algebraic equations",
      "algebraic geometry",
      "finite dimensional vector space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 44,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0804.4095K"
    }
  }
}