Most mathematics students learn how to solve systems of linear equations by Gaussian elimination, which is the procedure of converting a matrix to row echelon form. 

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But how about solving systems of equations given by polynomials in several variables, for example, the system

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x  +y  +z  = 2,

x  +y  +z  = 3,

x  +y  +z  = 4.

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One part of this project will study the method of Gröbner bases, which allows us to solve problems about polynomials in an algorithmic or computational fashion.  In particular, we will focus on how the size of a Gröbner basis grows when we create a family of systems of equations out of a given system. A good reference for Gröbner bases is chapter two of the book Ideals, Varieties, and Algorithms by Cox, Little, and O’Shea.

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Another part of the project will look at resolving (short for repeatedly solving) for dependencies among the equations in our system. This is also an algorithmic procedure with many interesting open problems regarding the possible numbers of dependence relations, called Betti numbers. We will explore what the possible Betti numbers are for some families of polynomial ideals. 

 

These questions can be approached both from a computational and from a theoretical point of view and we will take a combined approach. For the computational aspects, the students in this group will learn how to use the software system Macaulay2.