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.
| Identifer | oai:union.ndltd.org:csusb.edu/oai:scholarworks.lib.csusb.edu:etd-2028 |
| Date | 01 September 2019 |
| Creators | Redman, Lynn |
| Publisher | CSUSB ScholarWorks |
| Source Sets | California State University San Bernardino |
| Detected Language | English |
| Type | text |
| Format | application/pdf |
| Source | Electronic Theses, Projects, and Dissertations |
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