Undergraduate Texts in Mathematics are generally aimed at third-and fourthyear
undergraduate mathematics students at North American universities. These texts strive to …

In Chapter 1, we have seen how the algebra of the polynomial rings k [x1,..., xn] and the
geometry of affine algebraic varieties are linked. In this chapter, we will study the method of …

The world is continuous, but the mind is discrete. David Mumford We seek to bridge some
critical gaps between various? elds of mathematics by studying the interplay between the …

Enumerative combinatorics has undergone enormous development since the publication of
the first edition of this book in 1986. It has become more clear what are the essential topics …

It is undeniable that geometric ideas have been very important to the foundations of modern
discrete optimization. The influence that geometric algorithms have in optimization was …

Semi-linear sets, which are rational subsets of the monoid (Z^ d,+), have numerous
applications in theoretical computer science. Although semi-linear sets are usually given …

Computations with Barvinok's short rational generating functions are traditionally being
performed in the dual space, to avoid the combinatorial complexity of inclusion--exclusion …

We express the generating function for lattice points in a rational polyhedral cone with a
simplicial subdivision in terms of multivariate analogues of the h-polynomials of the …

We introduce variants of Barvinok's algorithm for counting lattice points in polyhedra. The
new algorithms are based on irrational signed decomposition in the primal space and the …

We present a polynomial time algorithm to compute any fixed number of the highest
coefficients of the Ehrhart quasi-polynomial of a rational simplex. Previously such algorithms …