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 …

This is a self-contained exposition of several core aspects of the theory of rational polyhedra
with a view towards algorithmic applications to efficient counting of integer points, a problem …

This paper starts by settling the computational complexity of the problem of integrating a
polynomial function $ f $ over a rational simplex. We prove that the problem is $\mathrm …

We are interested in the fast computation of the exact value of integrals of polynomial
functions over convex polyhedra. We present speed-ups and extensions of the algorithms …

Ehrhart discovered that the function that counts the number of lattice points in dilations of an
integral polytope is a polynomial. We call the coefficients of this polynomial Ehrhart …

Starting from the principle of locality in quantum field theory, which states that an object is
influenced directly only by its immediate surroundings, we review some features of the …

Quantitative information flow analysis (QIF) is a portfolio of software security assessment
techniques measuring the amount of confidential information leaked by a program to its …

A bstract We present a new formula for the biadjoint scalar tree amplitudes m (α| β) based on
the combinatorics of dual associahedra. Our construction makes essential use of the cones …

We equip the space of lattice cones with a coproduct which makes it a cograded,
coaugmented, connnected coalgebra. The exponential generating sum and exponential …