Monomials, convex bodies, and optimization.

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What is the least degree monomial of the form xᵃyᵇzᶜ that has at least two factors equal to x or y,  at least two factors equal to y or z, and at least two factors equal to z or x? It is xyz! We can arrive at this sort of question by doing algebra, that is by looking at monomial ideals and their powers. We can answer it by associating a geometric object called a polyhedron to the monomials we are interested in and finding the point of the polyhedron that has the smallest sum of its coordinates.

 

In the polymath REU project, we will explore polyhedra that can be constructed from monomial ideals and we will look for the optimal solutions to problems involving them.

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