Gröbner bases for toric ideals

Prerequisites: MAT220, MAT221, MAT224.

Description: For an ideal I in a polynomial ring S = k[x₁, ..., x_n], it is important to be able to make effective calculations with the ideal. E.g. to find the Hilbert polynomial S/I is to be able to say how large the ideal is, or to calculate I ∩ k[x₁, ..., x_r] where r < n (this is called elimination of variables), or to calculate the prime decomposition, or to calculate I : h.

The basic technique for all such calculations is to introduce a total ordering of the monomials. For every such ordering, one finds a specially selected set of generators for the ideal, called a Gröbner basis. Different monomial orderings give different Gröbner bases, and with the help of a suitable Gröbner basis, the above problems, and many others, can be solved effectively.

A class of ideals that have clear Gröbner bases is the class toric ideals. These are ideals that occur as kernels in algebra homomorphisms from k[x₁, ..., x_n] to another polynomial ring k[y₁, ..., y_m], where every x_i is mapped to a monomial x^a_1·x^a_2···x^a_n. We will study diverse classes of toric ideals and find Gröbner bases for these. In particular, we will be interested in whether all of the elements in the Gröbner basis have the lowest possible degree, i.e. 2.

References:

[1] D.Cox, B.Little, O'Shea, Ideals, Varieties and Algorithms, Springer Verlag.
[2] M.Kreuzer, L.Robbiano, Computational commutative algebra, Springer Verlag, 2000.
[3] B.Sturmfels, Gröbner bases and convex polytopes, University Lecture Series 8, AMS, 1996. (mr)