A combinatorial perspective on algebraic geometry

From combinatorial optimization to real algebraic geometry and back

In this paper, we explain the relations between combinatorial optimization and real algebraic geometry with a special focus to the quadratic assignment problem. We demonstrate how to write a quadratic optimization problem over discrete feasible set as a linear optimization problem over the cone of completely positive matrices. The latter formulation enables a hierarchy of approximations which r...

Proof of Two Combinatorial Results Arising in Algebraic Geometry

For a labeled tree on the vertex set [n] := {1, 2, . . . , n}, define the direction of each edge ij as i → j if i < j. The indegree sequence λ = 1122 . . . is then a partition of n−1. Let aλ be the number of trees on [n] with indegree sequence λ. In a recent paper (arXiv:0706.2049v2) Cotterill stumbled across the following two remarkable formulas aλ = (n− 1)! (n− k)!e1!(1!)1e2!(2!)2 . . . and

Enriques on Algebraic Geometry

tures of Bernoulli polynomials and gamma functions he has listed only the most important works. The bibliography is a very useful one. It is hardly to be expected that it should be complete. In fact I have found a considerable number of omissions by checking it against the partial bibliography which I have collected in an incidental way during the past fifteen years. It is natural to expect tha...

Introductory Geometry: Algebraic Geometry

Lectures on Logarithmic Algebraic Geometry