**UM Math Graduate Students Seminar**

Dr. Anton Dochtermann

* University of Miami *

**will present**

**Basics of Gröbner bases **

Friday, February 22, 2013, 2:00pm

Ungar Building Room 402

**Abstract:**Fix an ideal I in a polynomial ring. A basic question in commutative algebra is to determine whether a given polynomial belongs to I. If the polynomial ring has just one variable, this is easy since all ideals are generated by a single element; we can use long division. In general, a `Gröbner basis' for I is a set of generators of I that have nice algorithmic properties. The construction of a Gröbner basis allows for a *multivariate* version of the division algorithm, as well as a *nonlinear* generalization of Gaussian elimination for solving systems of equations. We will discuss the basic notions of Gröbner bases and sketch the famous algorithm of Buchberger, which shows that a Gröbner basis can be obtained from any generating set of I. Time permitting we will mention some connections to polyhedral and tropical geometry.

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