Convex bodies and algebraic equations on affine varieties
By Kiumars Kaveh, Askold Khovanskii. 2008
58 citations recorded by the source corpus.
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).
View Paper: https://www.semanticscholar.org/paper/468da10811569d802d7337ce6f5597de9bbaec15
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
- M. Mustaţă
- R. Lazarsfeld
- Chi Li
- Kiumars Kaveh
- S. Boucksom
- Tommaso de Fernex
- Steven Dale Cutkosky
- Mattias Jonsson
Citations in the corpus
- Convex Bodies Associated to Linear Series (2008, background): “Given such a vector ~a, it follows from the definitions that Γ(X)N~a = Γ(~aD) ⊆ Nd ×N~a,”