# Algebraic Methods for Proving Geometric Theorems

9-2019

Thesis

## Degree Name

Master of Arts in Mathematics

Mathematics

Wallace, Laura

## Abstract

Algebraic geometry is the study of systems of polynomial equations in one or more variables. Thinking of polynomials as functions reveals a close connection between affine varieties, which are geometric structures, and ideals, which are algebraic objects. An affine variety is a collection of tuples that represents the solutions to a system of equations. An ideal is a special subset of a ring and is what provides the tools to prove geometric theorems algebraically. In this thesis, we establish that a variety depends on the ideal generated by its defining equations. The ability to change the basis of an ideal without changing the variety is a powerful tool in determining a variety. In general, the quotient and remainder on division of polynomials in more than one variable are not unique. One property of a Groebner basis is that it yields a unique remainder on division.

To prove geometric theorems algebraically, we first express the hypotheses and conclusions as polynomials. Then, with the aid of a computer, apply the Groebner Basis Algorithm to determine if the conclusion polynomial(s) vanish on the same variety as the hypotheses.

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