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Algebra

2019

Polytopes

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Full-Text Articles in Physical Sciences and Mathematics

Dissertation_Davis.Pdf, Brian Davis Mar 2019

Dissertation_Davis.Pdf, Brian Davis

brian davis

Simplices are the ``simplest" examples of polytopes, and yet they exhibit much of the rich and subtle combinatorics and commutative algebra of their more general cousins. In this way they are sufficiently complicated --- insights gained from their study can inform broader research in Ehrhart theory and associated fields.

In this dissertation we consider two previously unstudied properties of lattice simplices; one algebraic and one combinatorial. The first is the Poincare series of the associated semigroup algebra, which is substantially more complicated than the Hilbert series of that same algebra. The second is the partial ordering of the elements of …

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Lattice Simplices: Sufficiently Complicated, Brian Davis Jan 2019

Lattice Simplices: Sufficiently Complicated, Brian Davis

Theses and Dissertations--Mathematics

Simplices are the "simplest" examples of polytopes, and yet they exhibit much of the rich and subtle combinatorics and commutative algebra of their more general cousins. In this way they are sufficiently complicated --- insights gained from their study can inform broader research in Ehrhart theory and associated fields.

In this dissertation we consider two previously unstudied properties of lattice simplices; one algebraic and one combinatorial. The first is the Poincar\'e series of the associated semigroup algebra, which is substantially more complicated than the Hilbert series of that same algebra. The second is the partial ordering of the elements of …

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