Lecture I:  Toric Ideals

-> The number of solutions to a system of sparse polynomial equations
   * Recall B\'ezout's Theorem
   * Example where B\'ezout is inadequate

-> Newton polytope as a multivariate generalization of degree.
   * General (unmixed) sparse system
   * Formulate as pullback of linear forms
   * Define the affine toric variety for a subset A of monomials

-> Discuss why the word toric
   * Do not discuss the construction from a fan
   * The two coincide when N A is saturated

-> Elementary questions about I_A, the ideal of a toric variety
   * Example, rational cubic, hexagon (Segre and Veronese in exercises)
   * Formulate ideal as the kernel of the map dual to the parametrization
               k[t_i^\pm]  <---- k[x_a|a \in A]
   * Theorem:  I_A is the linear span of the obvious binomials.
   * Dimension of X_A is the rank of A (via tangent spaces)
   * Same result, but for Krull dimension

-> Finer questions about I_A
   * Generated by vectors in the kernel of A
   * Binomial Groebner bases
   * Quadratic binomials and the conjecture.

-> Homogenize the ideal
   * Define I_A+
   * Example of Hexagon
   * The one bad triangle in R^2
   * Koelman's Theorem
   * BGT Theorem, and some open questions.