Fall 2015

Friday, August 28th, 2015

Organizational Meeting

In this organizational meeting we will discuss the seminar's plans for the 2015-2016 year. While the organizers have some ideas about where we would like the seminar to go this year, we are open to suggestions and will gladly go in different directions if people want to. We will hopefully assign talks for the Fall 2015 semester, so if you'd like to speak in the seminar this fall please attend!

Friday, October 2nd, 2015

Peter Thompson

Monomials and the Division Algorithm in k[x_1,...,x_n] and Groebner Bases

We generalize the polynomial division algorithm in k[x] to polynomials in k[x_1,...,x_n]. The presence of several variables creates ambiguities that were not present in the single-variable case. We introduce the notion of a monomial ordering and look at several examples. We discuss the need for Groebner bases and use them to solve the ideal membership problem.

Friday, October 9th, 2015

Peter Thompson

Buchberger's Algorithm

Groebner bases solve the ideal description problem, but the existence theorem is non-constructive. For practical purposes, we would like an algorithm that produces a Groebner basis for a polynomial ideal defined by any generating set. We describe such an algorithm and discuss various refinements.

Friday, October 16th, 2015

Eli Amzallag

The Elimination and Extension Theorems

Given a system of algebraic equations, one can ask if the elimination methods for linear systems studied in elementary linear algebra can be adapted to solve it. The hope would be that, (as in the linear case), the difficulty of solving the system is reduced by eliminating variables from some given equations or by producing equations that are consequences of the system and have fewer variables. If we then solve the equations with fewer variables or are handed solutions to those equations, we can now go "in reverse" and try to build up a full solution to the system from the partial solution, (as one does in solving linear systems by putting them in "triangular form" and then "back-substituting"). We will see when this idea "works" and how one can use Groebner bases to carry it out.

Friday, October 23rd, 2015

Eli Amzallag

Parametrized Varieties and Implicitization

We complete our discussion of the Extension Theorem and use the theorem to solve the implicitization problem. That is, given equations with some variables in terms of polynomial or rational expressions involving other variables (that we call parameters), we discuss when it is possible to eliminate the parameters and how this might be done. To this end, we will introduce a geometric version of the Extension Theorem. If time permits, we will use the example of the Whitney umbrella surface to illustrate these results.

Friday, October 30th, 2015

Richard Gustavson

Algorithms in Commutative Algebra

In order to work with examples in commutative algebra, we need algorithms that allow us to compute such things as a Groebner basis of an ideal or a basis for the intersection of two ideals, among other things. We have already seen examples of such algorithms, such as the division algorithm in a multivariate polynomial ring and Buchberger's algorithm for computing a Groebner basis of an ideal. In this talk, we will first present a new proof of Hilbert's Nullstellensatz which uses Groebner bases. We will then use the correspondence between ideals in multivariate polynomial rings and affine varieties to present many commutative algebraic algorithms, including an algorithm to determine if a multivariate polynomial is in the radical of a given ideal, and algorithms to compute the least common multiple and greatest common divisor of multivariate polynomials.

Friday, November 6th, 2015

Richard Gustavson

Algorithms in Commutative Algebra II: Ideal Quotients and Saturations

In this talk, we continue our discussion of algorithms in the ring of multivariate polynomial functions. We will focus on ideal quotients and saturation ideals, which are the algebraic analogue of the difference of two affine varieties. We will demonstrate the algebra-geometry correspondence between these concepts and describe algorithms to compute both the ideal quotient and the ideal saturation of two ideals.

Friday, November 13th, 2015

Eli Amzallag

Prime and Primary Decomposition

We study ideal decomposition in multivariate polynomial rings and prove the Lasker-Noether Theorem on minimal primary decomposition. We also discuss how these decompositions can be understood geometrically. Although we will not focus on constructive versions of these decompositions, if time permits we will sketch the ideas of some of the algorithms that return primary decomposition of an input ideal, specifically ones implemented in programs such as Macaulay2 and Maple.

Friday, November 20th, 2015

Richard Gustavson

The Closure Theorem

In this talk, we will prove the closure theorem, which states the relationship between the projection of a variety onto a subspace and the zero-set of the corresponding elimination ideal. We already proved the first part of the closure theorem, which says that the zero-set of the elimination ideal is the Zariski closure of the projection. In this talk, we will prove the second part, which determines what "else" is in the zero-set of the elimination ideal, besides the projection. This will require a lot of technical details; we will provide as many proofs as possible.

Friday, December 4th, 2015

Mengxiao Sun

Finiteness and Relative Finiteness

When we deal with a system of polynomial equations, we would like to know the number of solutions. Finiteness Theorem provides an algorithmic criterion to determine when a system of polynomial equations has only finitely many solutions in an algebraically closed field. In this talk, we will prove Finiteness Theorem and describe a method for finding explicit solutions to a system of polynomial equations in the field of complex numbers. We can generalize Finiteness Theorem to a relative version, which is the Relative Finiteness Theorem, if we allow the ideal of a polynomial ring k[x_1,…,x_n] to depend on a set of parameters y_1,…,y_m. If time permits, we will prove Relative Finiteness Theorem and introduce the geometric interpretation of it.

Friday, December 11th, 2015

Peter Thompson

Functions on Affine Varieties

We study the coordinate of an affine variety, that is, the ring of polynomial functions whose domain is that variety. There is an algebra-geometry dictionary between subvarieties and ideals of the coordinate ring. A polynomial map between varieties induces a homomorphism on their coordinate rings and a rational map between varieties induces a homomorphism on their function fields. This leads to two different notions of how two varieties can be considered the same: isomorphism and birational equivalence.