**Department of Mathematics**

**BILKENT **

**ALGEBRA SEMINAR**

**"****An application of graph matchings in algebraic geometry****"**

By

# ANARGYROS
KATSAMPEKIS

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**Abstract:**** Toric ideals are binomial ideals which
represent the algebraic relations of finite sets of power products. A basic
problem in Commutative Algebra asks one to compute the least number of
polynomials needed to generate the toric ideal up to
radical. This number is commonly known as the arithmetical rank of a toric ideal. Computing the arithmetical rank is one of the
classical problems of Algebraic Geometry which remains open even for very
simple cases. A usual approach to this problem is to restrict to a certain
class of polynomials and ask how many polynomials from this class can generate
the toric ideal up to radical. Restricting the
polynomials to the class of binomials we arrive at the notion of the binomial
arithmetical rank of a toric ideal. In the talk we
study the binomial arithmetical rank of the toric
ideal ***I*_{G}* ***of a finite graph ***G ***in two cases:**

**(1) ***G ***is bipartite,**

**(2) ***I*_{G}* ***is generated by quadratic
binomials.**

**Using a
generalized notion of a matching in a graph and circuits of toric
ideals, we prove that, in both cases, the binomial arithmetical rank equals the
minimal number of generators of ***I*_{G}**.**

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**Date: ****Monday, April 4, 2016**

**Time: ****14.40-15.30**

**Place: ****Mathematics Seminar Room,
SA-141**

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**All are most cordially invited.**

Tea and cookies: before the seminar

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