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Day: Wednesday, February 22
Time: 4:00pm
Place: BH 401
Title: Residues and Newton Polytopes
Speaker: Ivan Soprunov
Abstract: Given a Laurent polynomial f in n variables, its Newton polytope is
defined as the convex hull in R^n of the exponent vectors of the
monomials appearing in f. The theory of the Newton polytopes is
concerned with the study of algebraic sets in (C*)^n defined by a system
f_1=...=f_k=0 of Laurent polynomials with fixed Newton polytopes and
generic coefficients.
There is a striking connections between the theory of Newton polytopes
and toric varieties, discovered by Khovanskii in the mid 70's.
His discovery led to many beautiful results in both algebraic geometry
and polyhedral combinatorics (combinatorial Riemann-Roch theorem,
Stanley's theorem etc.)
One of the fundamental invariants of polynomial systems with finitely
many solutions is the global residue. It is a linear function on the
space of polynomials which has a rational expression in the coefficients
of the system. Global residues have numerous applications ranging from
elimination algorithms in commutative algebra to integral representation
formulae in complex analysis. I will present a new algorithm for
computing the global residue explicitly when the Newton polytopes of the
system are full-dimensional. This will bring us to interesting questions
about optimization and lattice point enumeration in Minkowski sums, and
sparse polynomial interpolation.
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