Date:
-
Location:
POT 745
Speaker(s) / Presenter(s):
Ivan Soprunov
Title: On zero dimensional complete intersections in the torus
Abstract: Consider an n-variate system of n Laurent polynomials over an algebraically closed field K with prescribed Newton polytopes P_1, ..., P_n. If the coefficients of the system are generic, the solution set Z consists of isolated points in the torus (K^*)^n. We concentrate on the following two questions. Given a polytope P, let L(P) be the space of Laurent polynomials spanned by monomials corresponding to the lattice points in P. What is the dimension of the subspace of those h\in L(P) that vanish on Z? If h\in L(P) does not vanish identically on Z, what is the smallest number of points p in Z where h(p)\neq 0? These questions are related to the multigraded Hilbert function of ideals in the homogeneous coordinate ring of a toric variety and the Cayley--Bacharach theorem. Although we cannot answer these questions in full generality, we will see how much can be said in terms of geometry of the polytopes P_1, ..., P_n and P. Both questions have applications to algebraic coding theory.
