March 21, Wednesday, 15:30, room 211

Evgeny Goncharov

Kouchnirenko’s theorem and its proof via Hilbert’s theorem

Kouchnirenko’s theorem states that the number of roots in the complex torus (С-0)^n of a generic system of equations P_1= …=P_n=0, where P_i’s are Laurent polynomials with a common Newton polytope \Delta is equal to n!Vol(\Delta). Askold Khovanskii has found around 15 different proofs of this theorem and we are going to review the one based on Hilbert’s Theorem. I also aim to provide some motivation and examples. A few general theorems of complex algebraic geometry will be used. If time permits I will also say something about a more elementary variation of the proof.