February 7, Wednesday, 15:30, room 211

Arina Arkhipova (HSE, UNIGE)

Mixed Fiber Polytopes

Let f_1=…=f_n=0 be a system of polynomial equations with the same Newton polytope N. The theorem proven by A. Kushnirenko allows us to compute the number of roots in (C\0)^n for the system f_1=…=f_n=0 provided that the coefficients of the latter are in general position. Namely, such a system has Vol(N) roots. The Bernstein–Kushnirenko theorem is a generalization of this result. It expresses the number of roots for a generic system with the Newton polytopes N_1,…,N_n that are not necessarily all the same. In this case, the answer is the so-called mixed volume of the polytopes N_1,…,N_n. The talk concerns the following generalization of the Bernstein-Kushnirenko theorem. Suppose X={f_1=…=f_{k}=0} is a complete intersection in (C\0)^n and p:(C\0)^n–>(C\0)^m is a projection. A.Esterov and A.Khovanskii proved that if the image p(X) in (C\0)^m is a hypersurface {g=0}, then the Newton polytope of g can be described in terms of the Newton polytopes of the polynomials defining X. Namely, if their Newton polytopes are all equal to N, then the answer is the so-called fiber polytope of N w.r.t. the projection p. If their Newton polytopes N_1,..,N_k are not all the same, then the answer is the mixed fiber polytope of N_1,..,N_k w.r.t. to the projection p. We will outline the proof of this result as well as existence, construction and some important properties of the (mixed) fiber bodies.

No prerequisites required: I will gently introduce all necessary notions during the talk and give many examples. The talk will consist of two parts with a short break in the middle.