Toric Euler-Jacobi theorem and its application to algebraic codes.
The Euler-Jacobi theorem is a powerful generalization of the classical residue formula from complex analysis to multivariate
polynomial systems. For example, Pascal's theorem on a hexagon inscribed in a conic, or Chasles' theorem on intersecting cubics are consequences of the Euler-Jacobi theorem.
The theory of Newton polytopes deals with multivariate polynomial systems that involve an additional ``discrete structure''. There is a striking connections between the theory of Newton polytopes and toric geometry, discovered by Khovanskii in the mid 70's.
His discovery led to many beautiful results in both algebraic geometry and polyhedral combinatorics. One of them is Khovanskii's ``toric" counterpart of the Euler-Jacobi theorem.
We will discuss a new application of the Toric Euler-Jacobi theorem to algebraic coding theory. In particular, we will obtain a generalization of earlier results of Gold-Little-Schenck and Ballico-Fontanari on evaluation codes on complete intersections in the projective space.