Date of Award

2012

Document Type

Honors Thesis (Open Access)

Department

Colby College. Mathematics and Statistics Dept.

Advisor(s)

Fernando Gouvêa

Second Advisor

Andreas Malmendier

Abstract

Puzzled by equations in multiple variables for centuries, mathematicians have made relatively few strides in solving these seemingly friendly, but unruly beasts. Currently, there is no systematic method for finding all rational values, that satisfy any equation with degree higher than a quadratic. This is bizarre. Solving these has preoccupied great minds since before the formal notion of an equation existed. Before any sort of mathematical formality, these questions were nested in plucky riddles and folded into folk tales. Because they are so simple to state, these equations are accessible to a very general audience. Yet an astounding amount of mathematical power is needed to even begin to generate universal results. On the one hand, it is easy to see that solutions do or do not exist for certain equations, but finding and proving the exact number of solutions is really hard, maybe impossible in some cases. On the other hand, this makes it a wonderful topic to research. The problems are beautiful and elegant to state, and accessible to anyone with some basic undergraduate knowledge. Yet, to even begin to solve these problems requires sophisticated tools from the far corners of geometry, topology, analysis and algebra.

Keywords

multiple variable equations

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