Polyhedral Theory and Commutative Algebra

An expository account is presented describing the use of methods of commutative algebra to solve problems concerning the enumeration of faces of convex polytopes. Assuming only basic knowledge of vector spaces and polynomial rings, the enumeration theory of Stanley is developed to the point where one can see how the Upper Bound Theorem for spheres is proved. A briefer account is then given of the extension of these techniques which yielded the proof of the necessity of McMullen’s conjectured characterization of the f-vectors of convex polytopes. The latter account includes a glimpse of the application of these methods to the study of integer solutions to systems of linear inequalities.

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