Abstract

I start with recalling the well-known theorem of Bernstein-Kushnirenko on the number of solutions of a system of Laurent polynomials in terms of volume of polytopes. This theorem plays an important role in the theory of Newton polytopes and toric geometry. Next, I discuss a recent development which generlaizes this to arbitrary varieties. We associate convex bodies to finite dimensional subsapces of rational functions (equivalently linear systems) on arbitrary algebraic varieties, far generalizing the notion of Newton polytope of a polynomial. We call them Newton-Okounkov bodies.
This is based on a basic construction of convex bodies for semigroups of integral points. This approach gives a nice result on the Hilbert function of a very general class of graded algebras. As corollaries we get elementary proofs and extensions of several important results in algebraic geometry, commutative algebra and convex geometry. It is worth mentioning that several well-known polytopes appearing in representation theory (of Lie groups) are special cases of these bodies. The methods and constructions can be applied to other situations e.g. arithmetic geometry. This is a joint work with A. G. Khovanskii and is based on previous works of A. Okounkov. The talk assumes minimum background and should be accessible to an audience with elementary knowledge in algebra and geometry. I expect the topics discussed to be of special interest for poeple in geometry and commutative algebra as well as combinatorics and representation theory.



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